Society for Industrial and Applied Mathematics: SIAM Journal on Mathematical Analysis: Table of Contents
Table of Contents for SIAM Journal on Mathematical Analysis. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/sjmaah?ai=s2&mi=3bfys9&af=R
Society for Industrial and Applied Mathematics: SIAM Journal on Mathematical Analysis: Table of Contents
Society for Industrial and Applied Mathematics
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SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjmaah/cover.jpg
https://epubs.siam.org/loi/sjmaah?ai=s2&mi=3bfys9&af=R
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Poisson Equation on Wasserstein Space and Diffusion Approximations for Multiscale McKean–Vlasov Equation
https://epubs.siam.org/doi/abs/10.1137/22M1536856?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1495-1524, April 2024. <br/> Abstract. We consider the fully-coupled McKean–Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. An extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1495-1524, April 2024. <br/> Abstract. We consider the fully-coupled McKean–Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. An extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.
Poisson Equation on Wasserstein Space and Diffusion Approximations for Multiscale McKean–Vlasov Equation
10.1137/22M1536856
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yun Li
Fuke Wu
Longjie Xie
Poisson Equation on Wasserstein Space and Diffusion Approximations for Multiscale McKean–Vlasov Equation
56
2
1495
1524
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1536856
https://epubs.siam.org/doi/abs/10.1137/22M1536856?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Modulated Energy Estimates for Singular Kernels and their Applications to Asymptotic Analyses for Kinetic Equations
https://epubs.siam.org/doi/abs/10.1137/22M1537643?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1525-1559, April 2024. <br/> Abstract. In this paper, we provide modulated interaction energy estimates for the kernel [math] with [math] and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified small inertia limit of kinetic equations with singular nonlocal interactions. The aggregation equations with singular interaction kernels are rigorously derived. We also study the rigorous quantified hydrodynamic limit of the kinetic equation to derive the isothermal Euler or pressureless Euler system with the nonlocal singular interaction forces.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1525-1559, April 2024. <br/> Abstract. In this paper, we provide modulated interaction energy estimates for the kernel [math] with [math] and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified small inertia limit of kinetic equations with singular nonlocal interactions. The aggregation equations with singular interaction kernels are rigorously derived. We also study the rigorous quantified hydrodynamic limit of the kinetic equation to derive the isothermal Euler or pressureless Euler system with the nonlocal singular interaction forces.
Modulated Energy Estimates for Singular Kernels and their Applications to Asymptotic Analyses for Kinetic Equations
10.1137/22M1537643
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Young-Pil Choi
Jinwook Jung
Modulated Energy Estimates for Singular Kernels and their Applications to Asymptotic Analyses for Kinetic Equations
56
2
1525
1559
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1537643
https://epubs.siam.org/doi/abs/10.1137/22M1537643?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Fuglede-Type Arguments for Isoperimetric Problems and Applications to Stability Among Convex Shapes
https://epubs.siam.org/doi/abs/10.1137/23M1567412?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1560-1603, April 2024. <br/> Abstract.This paper is concerned with the stability of the ball for a class of isoperimetric problems under a convexity constraint. Considering the problem of minimizing [math] among convex subsets of [math] of fixed volume, where [math] is the perimeter functional, [math] is a perturbative term, and [math] is a small parameter, the stability of the ball for this perturbed isoperimetric problem means that the ball is the unique (local, up to translation) minimizer for any [math] sufficiently small. We investigate independently two specific cases where [math] is an energy arising from PDE theory, namely, the capacity and the first Dirichlet eigenvalue of a domain [math]. While in both cases stability fails among all shapes, in the first case we prove the (nonsharp) stability of the ball among convex shapes, by building an appropriate competitor for the capacity of a perturbation of the ball. In the second case we prove the sharp stability of the ball among convex shapes by providing the optimal range of [math] such that stability holds, relying on the selection principle technique and a regularity theory under convexity constraint.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1560-1603, April 2024. <br/> Abstract.This paper is concerned with the stability of the ball for a class of isoperimetric problems under a convexity constraint. Considering the problem of minimizing [math] among convex subsets of [math] of fixed volume, where [math] is the perimeter functional, [math] is a perturbative term, and [math] is a small parameter, the stability of the ball for this perturbed isoperimetric problem means that the ball is the unique (local, up to translation) minimizer for any [math] sufficiently small. We investigate independently two specific cases where [math] is an energy arising from PDE theory, namely, the capacity and the first Dirichlet eigenvalue of a domain [math]. While in both cases stability fails among all shapes, in the first case we prove the (nonsharp) stability of the ball among convex shapes, by building an appropriate competitor for the capacity of a perturbation of the ball. In the second case we prove the sharp stability of the ball among convex shapes by providing the optimal range of [math] such that stability holds, relying on the selection principle technique and a regularity theory under convexity constraint.
Fuglede-Type Arguments for Isoperimetric Problems and Applications to Stability Among Convex Shapes
10.1137/23M1567412
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Raphaël Prunier
Fuglede-Type Arguments for Isoperimetric Problems and Applications to Stability Among Convex Shapes
56
2
1560
1603
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1567412
https://epubs.siam.org/doi/abs/10.1137/23M1567412?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Torsional Rigidity in Random Walk Spaces
https://epubs.siam.org/doi/abs/10.1137/23M1553200?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1604-1642, April 2024. <br/> Abstract. In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set [math] with the spectral [math]-heat content of [math], which gives rise to a complete description of the nonlocal torsional rigidity of [math] by using uniquely probability terms involving the set [math], and we recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability terms. For the random walk in [math] associated with a nonsingular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling, we recover the classical Saint-Venant inequality. We also study the nonlocal [math]-torsional rigidity and its relation with the nonlocal Cheeger constants, and we prove a nonlocal version of the Pólya–Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1604-1642, April 2024. <br/> Abstract. In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set [math] with the spectral [math]-heat content of [math], which gives rise to a complete description of the nonlocal torsional rigidity of [math] by using uniquely probability terms involving the set [math], and we recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability terms. For the random walk in [math] associated with a nonsingular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling, we recover the classical Saint-Venant inequality. We also study the nonlocal [math]-torsional rigidity and its relation with the nonlocal Cheeger constants, and we prove a nonlocal version of the Pólya–Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.
Torsional Rigidity in Random Walk Spaces
10.1137/23M1553200
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
José M. Mazón
Julián Toledo
Torsional Rigidity in Random Walk Spaces
56
2
1604
1642
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1553200
https://epubs.siam.org/doi/abs/10.1137/23M1553200?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Evolution of Dispersal in Advective Homogeneous Environments: Inflow Versus Outflow
https://epubs.siam.org/doi/abs/10.1137/22M1539095?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1643-1671, April 2024. <br/> Abstract. We consider a single species model and a two species competition model in one-dimensional open advective environments featured by an inflow (resp., outflow) of individuals at the upstream (resp., downstream) end as measured by a parameter [math] (resp., [math]). The two species are assumed to follow the same population dynamics but have different random diffusion rates. Under certain mild conditions on [math] and [math], we first determine clearly the global dynamics of the single species model in terms of critical habitat size or critical advection speed, and then further give a complete understanding on the global dynamics of the two species competition model. Our results suggest that in an open environment with mild inflow and outflow rates and with totally unfavorable boundary effect (outflow rate greater than inflow rate), “faster diffusion can evolve,” a different mechanism behind evolution of dispersal from that (“slower diffusion can evolve”) observed by Tang and Chen [J. Differential Equations, 269 (2020), pp. 1465–1483]. We also discuss other scenarios with different settings of [math] and [math] and propose problems that deserve future investigation.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1643-1671, April 2024. <br/> Abstract. We consider a single species model and a two species competition model in one-dimensional open advective environments featured by an inflow (resp., outflow) of individuals at the upstream (resp., downstream) end as measured by a parameter [math] (resp., [math]). The two species are assumed to follow the same population dynamics but have different random diffusion rates. Under certain mild conditions on [math] and [math], we first determine clearly the global dynamics of the single species model in terms of critical habitat size or critical advection speed, and then further give a complete understanding on the global dynamics of the two species competition model. Our results suggest that in an open environment with mild inflow and outflow rates and with totally unfavorable boundary effect (outflow rate greater than inflow rate), “faster diffusion can evolve,” a different mechanism behind evolution of dispersal from that (“slower diffusion can evolve”) observed by Tang and Chen [J. Differential Equations, 269 (2020), pp. 1465–1483]. We also discuss other scenarios with different settings of [math] and [math] and propose problems that deserve future investigation.
Evolution of Dispersal in Advective Homogeneous Environments: Inflow Versus Outflow
10.1137/22M1539095
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yin Wang
Qingxiang Xu
Peng Zhou
Evolution of Dispersal in Advective Homogeneous Environments: Inflow Versus Outflow
56
2
1643
1671
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1539095
https://epubs.siam.org/doi/abs/10.1137/22M1539095?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Global Dynamics to the Periodic Ferromagnetic Spin Chain System
https://epubs.siam.org/doi/abs/10.1137/22M1531038?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1672-1726, April 2024. <br/> Abstract. In this paper, we study the dynamical system driven by the ferromagnetic spin chain system defined in [math] with or without external fields. In the first part, we prove the convergence to equilibriums for arbitrary large initial data of finite energy and prove the existence of heteroclinic orbits emerging from equilibriums of nontrivial energy. Moreover, we find a topological property of these heteroclinic orbits. In the second part, we study the stabilization of these equilibriums by adding an external field. Particularly, we prove that any given finite numbers of equilibriums can be stabilized simultaneously by choosing some external field.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1672-1726, April 2024. <br/> Abstract. In this paper, we study the dynamical system driven by the ferromagnetic spin chain system defined in [math] with or without external fields. In the first part, we prove the convergence to equilibriums for arbitrary large initial data of finite energy and prove the existence of heteroclinic orbits emerging from equilibriums of nontrivial energy. Moreover, we find a topological property of these heteroclinic orbits. In the second part, we study the stabilization of these equilibriums by adding an external field. Particularly, we prove that any given finite numbers of equilibriums can be stabilized simultaneously by choosing some external field.
Global Dynamics to the Periodic Ferromagnetic Spin Chain System
10.1137/22M1531038
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Li Ze
Changzheng Qu
Global Dynamics to the Periodic Ferromagnetic Spin Chain System
56
2
1672
1726
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1531038
https://epubs.siam.org/doi/abs/10.1137/22M1531038?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Minimal Mass Blow-Up Solutions for the [math]-Critical NLS with the Delta Potential for Even Data in One Dimension
https://epubs.siam.org/doi/abs/10.1137/23M1566091?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1727-1769, April 2024. <br/> Abstract. We consider the [math]-critical nonlinear Schrödinger equation (NLS) with the delta potential [math] where [math] and [math] is the Dirac delta distribution at [math]. Local well-posedness theory, together with the sharp Gagliardo–Nirenberg inequality and the conservation laws of mass and energy, implies that the solution with mass less than [math] is global existence in [math], where [math] is the ground state of the [math]-critical NLS without the delta potential (i.e., [math]). We are interested in the dynamics of the solution with threshold mass [math] in [math]. First, for the case [math], such a blow-up solution exists due to the pseudoconformal symmetry of the equation and is unique up to the symmetries of the equation in [math] from Merle [Duke Math. J., 69 (1993), pp. 427–454] and recently in [math] from Dodson [arXiv:2104.11690, 2021]. Second, for the case [math], a simple variational argument with the conservation laws of mass and energy implies that even solutions with threshold mass exist globally in [math]. Finally, for the case [math], we show the existence of even threshold solutions with blow-up speed determined by the sign (i.e., [math]) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense. The key ingredients here, including the Energy–Morawetz argument and the compactness method as well as modulation analysis, are close to the original one in Raphaël and Szeftel [J. Amer. Math. Soc., 24 (2011), pp. 471–546].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1727-1769, April 2024. <br/> Abstract. We consider the [math]-critical nonlinear Schrödinger equation (NLS) with the delta potential [math] where [math] and [math] is the Dirac delta distribution at [math]. Local well-posedness theory, together with the sharp Gagliardo–Nirenberg inequality and the conservation laws of mass and energy, implies that the solution with mass less than [math] is global existence in [math], where [math] is the ground state of the [math]-critical NLS without the delta potential (i.e., [math]). We are interested in the dynamics of the solution with threshold mass [math] in [math]. First, for the case [math], such a blow-up solution exists due to the pseudoconformal symmetry of the equation and is unique up to the symmetries of the equation in [math] from Merle [Duke Math. J., 69 (1993), pp. 427–454] and recently in [math] from Dodson [arXiv:2104.11690, 2021]. Second, for the case [math], a simple variational argument with the conservation laws of mass and energy implies that even solutions with threshold mass exist globally in [math]. Finally, for the case [math], we show the existence of even threshold solutions with blow-up speed determined by the sign (i.e., [math]) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense. The key ingredients here, including the Energy–Morawetz argument and the compactness method as well as modulation analysis, are close to the original one in Raphaël and Szeftel [J. Amer. Math. Soc., 24 (2011), pp. 471–546].
Minimal Mass Blow-Up Solutions for the [math]-Critical NLS with the Delta Potential for Even Data in One Dimension
10.1137/23M1566091
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Xingdong Tang
Guixiang Xu
Minimal Mass Blow-Up Solutions for the [math]-Critical NLS with the Delta Potential for Even Data in One Dimension
56
2
1727
1769
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1566091
https://epubs.siam.org/doi/abs/10.1137/23M1566091?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Asymptotic Expansion of the Spectrum for Periodic Schrödinger Operators
https://epubs.siam.org/doi/abs/10.1137/22M1526228?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1770-1808, April 2024. <br/> Abstract. We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schrödinger-type operator with a confining potential and with the principle part of a periodic elliptic operator in divergence form. We compare the spectrum to the homogenized operator and characterize the corrections up to arbitrarily high order.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1770-1808, April 2024. <br/> Abstract. We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schrödinger-type operator with a confining potential and with the principle part of a periodic elliptic operator in divergence form. We compare the spectrum to the homogenized operator and characterize the corrections up to arbitrarily high order.
Asymptotic Expansion of the Spectrum for Periodic Schrödinger Operators
10.1137/22M1526228
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Scott Armstrong
Raghavendra Venkatraman
Asymptotic Expansion of the Spectrum for Periodic Schrödinger Operators
56
2
1770
1808
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1526228
https://epubs.siam.org/doi/abs/10.1137/22M1526228?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Initial-Boundary Value Problems for Poiseuille Flow of Nematic Liquid Crystal via Full Ericksen–Leslie Model
https://epubs.siam.org/doi/abs/10.1137/23M1574567?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1809-1850, April 2024. <br/> Abstract. In this paper, we study the initial-boundary value problem for the Poiseuille flow of a hyperbolic-parabolic Ericksen–Leslie model of nematic liquid crystals in one space dimension. We consider a simplified system by restricting the Leslie coefficients to special cases such that some quantities are constants. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of a Hölder continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1809-1850, April 2024. <br/> Abstract. In this paper, we study the initial-boundary value problem for the Poiseuille flow of a hyperbolic-parabolic Ericksen–Leslie model of nematic liquid crystals in one space dimension. We consider a simplified system by restricting the Leslie coefficients to special cases such that some quantities are constants. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of a Hölder continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.
Initial-Boundary Value Problems for Poiseuille Flow of Nematic Liquid Crystal via Full Ericksen–Leslie Model
10.1137/23M1574567
SIAM Journal on Mathematical Analysis
2024-03-01T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Geng Chen
Yanbo Hu
Qingtian Zhang
Initial-Boundary Value Problems for Poiseuille Flow of Nematic Liquid Crystal via Full Ericksen–Leslie Model
56
2
1809
1850
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1574567
https://epubs.siam.org/doi/abs/10.1137/23M1574567?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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On the Global Well-Posedness for the Periodic Quintic Nonlinear Schrödinger Equation
https://epubs.siam.org/doi/abs/10.1137/22M1531063?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1851-1902, April 2024. <br/> Abstract. In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schrödinger equation on [math] with general data in the critical Sobolev space [math]. We show that if a solution remains bounded in [math] in its maximal interval of existence, then the solution exists globally in time.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1851-1902, April 2024. <br/> Abstract. In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schrödinger equation on [math] with general data in the critical Sobolev space [math]. We show that if a solution remains bounded in [math] in its maximal interval of existence, then the solution exists globally in time.
On the Global Well-Posedness for the Periodic Quintic Nonlinear Schrödinger Equation
10.1137/22M1531063
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Xueying Yu
Haitian Yue
On the Global Well-Posedness for the Periodic Quintic Nonlinear Schrödinger Equation
56
2
1851
1902
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1531063
https://epubs.siam.org/doi/abs/10.1137/22M1531063?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Nonlocal Bounded Variations with Applications
https://epubs.siam.org/doi/abs/10.1137/22M1520876?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1903-1935, April 2024. <br/> Abstract. Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ([math])-type spaces. Two different natural fractional analogs of classical [math] are considered: [math], a space induced from the Riesz-fractional gradient that has been recently studied by Comi–Stefani; and [math], induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics—this one is naturally related to the Caffarelli–Roquejoffre–Savin fractional perimeter. Our main theoretical result is that the latter [math] actually corresponds to the Gagliardo–Slobodeckij space [math]. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel predual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1903-1935, April 2024. <br/> Abstract. Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ([math])-type spaces. Two different natural fractional analogs of classical [math] are considered: [math], a space induced from the Riesz-fractional gradient that has been recently studied by Comi–Stefani; and [math], induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics—this one is naturally related to the Caffarelli–Roquejoffre–Savin fractional perimeter. Our main theoretical result is that the latter [math] actually corresponds to the Gagliardo–Slobodeckij space [math]. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel predual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.
Nonlocal Bounded Variations with Applications
10.1137/22M1520876
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Harbir Antil
Hugo Díaz
Tian Jing
Armin Schikorra
Nonlocal Bounded Variations with Applications
56
2
1903
1935
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1520876
https://epubs.siam.org/doi/abs/10.1137/22M1520876?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Almost Global Existence for Kirchhoff Equations Around Global Solutions
https://epubs.siam.org/doi/abs/10.1137/23M1556393?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1936-1958, April 2024. <br/> Abstract. It is well-known that the life span of solutions to Kirchhoff equations tends to infinity when initial data tend to zero. These results are usually referred to as almost global existence, at least in a neighborhood of the null solution. Here we extend this result by showing that the life span of solutions is lower semicontinuous, and in particular it tends to infinity whenever initial data tend to some limiting datum that originates a global solution. We also provide an estimate from below for the life span of solutions when initial data are close to some of the classes of data for which global existence is known, namely data with finitely many Fourier modes, analytic data, and quasi-analytic data.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1936-1958, April 2024. <br/> Abstract. It is well-known that the life span of solutions to Kirchhoff equations tends to infinity when initial data tend to zero. These results are usually referred to as almost global existence, at least in a neighborhood of the null solution. Here we extend this result by showing that the life span of solutions is lower semicontinuous, and in particular it tends to infinity whenever initial data tend to some limiting datum that originates a global solution. We also provide an estimate from below for the life span of solutions when initial data are close to some of the classes of data for which global existence is known, namely data with finitely many Fourier modes, analytic data, and quasi-analytic data.
Almost Global Existence for Kirchhoff Equations Around Global Solutions
10.1137/23M1556393
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Marina Ghisi
Massimo Gobbino
Almost Global Existence for Kirchhoff Equations Around Global Solutions
56
2
1936
1958
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1556393
https://epubs.siam.org/doi/abs/10.1137/23M1556393?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
-
Inverse Scattering for the Biharmonic Wave Equation with a Random Potential
https://epubs.siam.org/doi/abs/10.1137/22M1538399?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1959-1995, April 2024. <br/> Abstract. We consider the inverse random potential scattering problem for the two- and three-dimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials, and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the backscattering data averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1959-1995, April 2024. <br/> Abstract. We consider the inverse random potential scattering problem for the two- and three-dimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials, and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the backscattering data averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless.
Inverse Scattering for the Biharmonic Wave Equation with a Random Potential
10.1137/22M1538399
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Peijun Li
Xu Wang
Inverse Scattering for the Biharmonic Wave Equation with a Random Potential
56
2
1959
1995
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1538399
https://epubs.siam.org/doi/abs/10.1137/22M1538399?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
-
Steady States with Jump Discontinuity in a Receptor-Based Model with Hysteresis in Higher-Dimensional Domains
https://epubs.siam.org/doi/abs/10.1137/22M1509059?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 1996-2033, April 2024. <br/> Abstract. This paper deals with a receptor-based model which arises from the modeling of interactions between intracellular processes and diffusible signaling factors. We prove the existence of stationary solutions with jump discontinuity by a variational method. Then a singular perturbation problem with discontinuous nonlinearity is studied based on the patching argument and the implicit function theorem, to obtain a solution with a single transition layer. Moreover, we derive a sufficient condition for the stability of stationary solutions with jump discontinuity. In addition, some numerical simulations are presented to illustrate the theoretical results.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 1996-2033, April 2024. <br/> Abstract. This paper deals with a receptor-based model which arises from the modeling of interactions between intracellular processes and diffusible signaling factors. We prove the existence of stationary solutions with jump discontinuity by a variational method. Then a singular perturbation problem with discontinuous nonlinearity is studied based on the patching argument and the implicit function theorem, to obtain a solution with a single transition layer. Moreover, we derive a sufficient condition for the stability of stationary solutions with jump discontinuity. In addition, some numerical simulations are presented to illustrate the theoretical results.
Steady States with Jump Discontinuity in a Receptor-Based Model with Hysteresis in Higher-Dimensional Domains
10.1137/22M1509059
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Goro Akagi
Izumi Takagi
Conghui Zhang
Steady States with Jump Discontinuity in a Receptor-Based Model with Hysteresis in Higher-Dimensional Domains
56
2
1996
2033
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1509059
https://epubs.siam.org/doi/abs/10.1137/22M1509059?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona–Malik Type Equations
https://epubs.siam.org/doi/abs/10.1137/23M1569873?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2034-2062, April 2024. <br/> Abstract. We consider generalized solutions of the Perona–Malik equation in dimension one, defined as all possible limits of solutions to the semidiscrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times and, in particular, many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely, the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub-/super-solutions. The monotonicity results are proved for a more general class of evolution curves, that we call [math]-evolutions.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2034-2062, April 2024. <br/> Abstract. We consider generalized solutions of the Perona–Malik equation in dimension one, defined as all possible limits of solutions to the semidiscrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times and, in particular, many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely, the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub-/super-solutions. The monotonicity results are proved for a more general class of evolution curves, that we call [math]-evolutions.
Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona–Malik Type Equations
10.1137/23M1569873
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Massimo Gobbino
Nicola Picenni
Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona–Malik Type Equations
56
2
2034
2062
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1569873
https://epubs.siam.org/doi/abs/10.1137/23M1569873?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
-
Analysis of a Dilute Polymer Model with a Time-Fractional Derivative
https://epubs.siam.org/doi/abs/10.1137/23M1590767?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2063-2089, April 2024. <br/> Abstract. We investigate the well-posedness of a coupled Navier–Stokes–Fokker–Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modeled by a stochastic process exhibiting power-law waiting time in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modeled by a finitely extensible nonlinear elastic dumbbell model, and the drag term in the Fokker–Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order [math] and derive an energy inequality satisfied by weak solutions.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2063-2089, April 2024. <br/> Abstract. We investigate the well-posedness of a coupled Navier–Stokes–Fokker–Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modeled by a stochastic process exhibiting power-law waiting time in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modeled by a finitely extensible nonlinear elastic dumbbell model, and the drag term in the Fokker–Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order [math] and derive an energy inequality satisfied by weak solutions.
Analysis of a Dilute Polymer Model with a Time-Fractional Derivative
10.1137/23M1590767
SIAM Journal on Mathematical Analysis
2024-03-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Marvin Fritz
Endre Süli
Barbara Wohlmuth
Analysis of a Dilute Polymer Model with a Time-Fractional Derivative
56
2
2063
2089
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1590767
https://epubs.siam.org/doi/abs/10.1137/23M1590767?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model
https://epubs.siam.org/doi/abs/10.1137/23M1575573?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2090-2114, April 2024. <br/> Abstract. In recent years, there has been a spike in interest in multiphase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke’s law, Brinkman’s law, or Darcy’s law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth of results in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumor model (of Brinkman type) and an inviscid tumor model (of Darcy type). Specifically, we prove the convergence of solutions of the Brinkman nonlocal transport system toward a weak solution of the Darcy nonlinear parabolic system in the limit of vanishing viscosity.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2090-2114, April 2024. <br/> Abstract. In recent years, there has been a spike in interest in multiphase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke’s law, Brinkman’s law, or Darcy’s law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth of results in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumor model (of Brinkman type) and an inviscid tumor model (of Darcy type). Specifically, we prove the convergence of solutions of the Brinkman nonlocal transport system toward a weak solution of the Darcy nonlinear parabolic system in the limit of vanishing viscosity.
A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model
10.1137/23M1575573
SIAM Journal on Mathematical Analysis
2024-03-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Noemi David
Tomasz Dębiec
Mainak Mandal
Markus Schmidtchen
A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model
56
2
2090
2114
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1575573
https://epubs.siam.org/doi/abs/10.1137/23M1575573?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Inverse Resonance Problems for Energy-Dependent Potentials on the Half-Line
https://epubs.siam.org/doi/abs/10.1137/23M1591104?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2115-2148, April 2024. <br/> Abstract. We consider Schrödinger equations with energy-dependent potentials that are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schrödinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of isoresonance potentials and boundary condition parameters. Our strategy consists of exploiting a correspondence between Schrödinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for a Schrödinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2115-2148, April 2024. <br/> Abstract. We consider Schrödinger equations with energy-dependent potentials that are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schrödinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of isoresonance potentials and boundary condition parameters. Our strategy consists of exploiting a correspondence between Schrödinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for a Schrödinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition.
Inverse Resonance Problems for Energy-Dependent Potentials on the Half-Line
10.1137/23M1591104
SIAM Journal on Mathematical Analysis
2024-03-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Evgeny Korotyaev
Andrea Mantile
Dmitrii Mokeev
Inverse Resonance Problems for Energy-Dependent Potentials on the Half-Line
56
2
2115
2148
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1591104
https://epubs.siam.org/doi/abs/10.1137/23M1591104?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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The Scattering Resonances for Schrödinger-Type Operators with Unbounded Potentials
https://epubs.siam.org/doi/abs/10.1137/22M1498619?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2149-2170, April 2024. <br/> Abstract. This paper addresses the meromorphic continuation of the outgoing resolvent associated with Schrödinger-type operators in three dimensions. The first part focuses on the classical Schrödinger-type operator involving unbounded potentials. The absence of nonzero real poles for the outgoing resolvent is investigated. The second part examines the fractional Schrödinger operator, including both bounded and unbounded potentials. The analysis relies on a resolvent identity that establishes a connection between the resolvents of the fractional Schrödinger operator and its classical counterpart.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2149-2170, April 2024. <br/> Abstract. This paper addresses the meromorphic continuation of the outgoing resolvent associated with Schrödinger-type operators in three dimensions. The first part focuses on the classical Schrödinger-type operator involving unbounded potentials. The absence of nonzero real poles for the outgoing resolvent is investigated. The second part examines the fractional Schrödinger operator, including both bounded and unbounded potentials. The analysis relies on a resolvent identity that establishes a connection between the resolvents of the fractional Schrödinger operator and its classical counterpart.
The Scattering Resonances for Schrödinger-Type Operators with Unbounded Potentials
10.1137/22M1498619
SIAM Journal on Mathematical Analysis
2024-03-07T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Peijun Li
Xiaohua Yao
Yue Zhao
The Scattering Resonances for Schrödinger-Type Operators with Unbounded Potentials
56
2
2149
2170
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1498619
https://epubs.siam.org/doi/abs/10.1137/22M1498619?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Square Root Normal Fields for Lipschitz Surfaces and the Wasserstein Fisher Rao Metric
https://epubs.siam.org/doi/abs/10.1137/22M1544452?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2171-2190, April 2024. <br/> Abstract. The square root normal field (SRNF) framework is a method in the area of shape analysis that defines a (pseudo)distance between unparametrized surfaces. For piecewise linear surfaces it was recently proved that the SRNF distance between unparametrized surfaces is equivalent to the Wasserstein Fisher Rao (WFR) metric on the space of finitely supported measures on [math]. In the present article we extend this point of view to a much larger set of surfaces; we show that the SRNF distance on the space of Lipschitz surfaces is equivalent to the WFR distance between Borel measures on [math]. For the space of spherical surfaces this result directly allows us to characterize the noninjectivity and the (closure of the) image of the SRNF transform. In the last part of the paper we further generalize this result by showing that the WFR metric for general measure spaces can be interpreted as an optimization problem over the diffeomorphism group of an independent background space.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2171-2190, April 2024. <br/> Abstract. The square root normal field (SRNF) framework is a method in the area of shape analysis that defines a (pseudo)distance between unparametrized surfaces. For piecewise linear surfaces it was recently proved that the SRNF distance between unparametrized surfaces is equivalent to the Wasserstein Fisher Rao (WFR) metric on the space of finitely supported measures on [math]. In the present article we extend this point of view to a much larger set of surfaces; we show that the SRNF distance on the space of Lipschitz surfaces is equivalent to the WFR distance between Borel measures on [math]. For the space of spherical surfaces this result directly allows us to characterize the noninjectivity and the (closure of the) image of the SRNF transform. In the last part of the paper we further generalize this result by showing that the WFR metric for general measure spaces can be interpreted as an optimization problem over the diffeomorphism group of an independent background space.
Square Root Normal Fields for Lipschitz Surfaces and the Wasserstein Fisher Rao Metric
10.1137/22M1544452
SIAM Journal on Mathematical Analysis
2024-03-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Emmanuel Hartman
Martin Bauer
Eric Klassen
Square Root Normal Fields for Lipschitz Surfaces and the Wasserstein Fisher Rao Metric
56
2
2171
2190
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1544452
https://epubs.siam.org/doi/abs/10.1137/22M1544452?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Blow-Up vs. Global Existence for a Fujita-Type Heat Exchanger System
https://epubs.siam.org/doi/abs/10.1137/23M1587440?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2191-2212, April 2024. <br/> Abstract. We analyze a reaction-diffusion system on [math] which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2191-2212, April 2024. <br/> Abstract. We analyze a reaction-diffusion system on [math] which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.
Blow-Up vs. Global Existence for a Fujita-Type Heat Exchanger System
10.1137/23M1587440
SIAM Journal on Mathematical Analysis
2024-03-13T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Samuel Tréton
Blow-Up vs. Global Existence for a Fujita-Type Heat Exchanger System
56
2
2191
2212
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1587440
https://epubs.siam.org/doi/abs/10.1137/23M1587440?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Existence and Stability of Dissipative Measure-Valued Solutions to the Full Compressible Magnetohydrodynamic Flows
https://epubs.siam.org/doi/abs/10.1137/22M1544397?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2213-2247, April 2024. <br/> Abstract.In this paper, we are concerned with dissipative measure-valued (DMV) solutions to the full compressible magnetohydrodynamics. The existence of the DMV solutions is established. Moreover, we prove that the strong solutions are stable in this class of generalized solution. Specifically, we demonstrate that a DMV solution is the same as the strong solution if they have the same initial data.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2213-2247, April 2024. <br/> Abstract.In this paper, we are concerned with dissipative measure-valued (DMV) solutions to the full compressible magnetohydrodynamics. The existence of the DMV solutions is established. Moreover, we prove that the strong solutions are stable in this class of generalized solution. Specifically, we demonstrate that a DMV solution is the same as the strong solution if they have the same initial data.
Existence and Stability of Dissipative Measure-Valued Solutions to the Full Compressible Magnetohydrodynamic Flows
10.1137/22M1544397
SIAM Journal on Mathematical Analysis
2024-03-13T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Bingkang Huang
Existence and Stability of Dissipative Measure-Valued Solutions to the Full Compressible Magnetohydrodynamic Flows
56
2
2213
2247
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1544397
https://epubs.siam.org/doi/abs/10.1137/22M1544397?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations
https://epubs.siam.org/doi/abs/10.1137/23M1563141?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2248-2285, April 2024. <br/> Abstract. In this paper we establish a sharp nonuniqueness result for stochastic [math]-dimensional ([math]) incompressible Navier–Stokes equations. First, for every divergence-free initial condition in [math] we show existence of infinitly many global-in-time probabilistically strong and analytically weak solutions in the class [math] for any [math]. Second, we prove that the above result is sharp in the sense that pathwise uniqueness holds in the class of [math] for some [math] such that [math], which is a stochastic version of Ladyzhenskaya–Prodi–Serrin criteria. Moreover, for the stochastic [math]-dimensional incompressible Euler equation, the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in Hoffmanová, Zhu, and Zhu [J. Eur. Math. Soc. (JEMS), to appear; Ann. Probab., 51 (2023), pp. 524–579], we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during convex integration scheme and construct directly solutions on the whole time interval [math].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2248-2285, April 2024. <br/> Abstract. In this paper we establish a sharp nonuniqueness result for stochastic [math]-dimensional ([math]) incompressible Navier–Stokes equations. First, for every divergence-free initial condition in [math] we show existence of infinitly many global-in-time probabilistically strong and analytically weak solutions in the class [math] for any [math]. Second, we prove that the above result is sharp in the sense that pathwise uniqueness holds in the class of [math] for some [math] such that [math], which is a stochastic version of Ladyzhenskaya–Prodi–Serrin criteria. Moreover, for the stochastic [math]-dimensional incompressible Euler equation, the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in Hoffmanová, Zhu, and Zhu [J. Eur. Math. Soc. (JEMS), to appear; Ann. Probab., 51 (2023), pp. 524–579], we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during convex integration scheme and construct directly solutions on the whole time interval [math].
Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations
10.1137/23M1563141
SIAM Journal on Mathematical Analysis
2024-03-14T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Weiquan Chen
Zhao Dong
Xiangchan Zhu
Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations
56
2
2248
2285
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1563141
https://epubs.siam.org/doi/abs/10.1137/23M1563141?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers
https://epubs.siam.org/doi/abs/10.1137/22M1525818?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2286-2311, April 2024. <br/> Abstract. We study vortex patches for the two-dimensional incompressible Euler equations. Prior works on this problem take the support of the vorticity (i.e., the vortex patch) to be a bounded region. We instead consider the horizontally periodic setting. This includes both the case of a periodic array of bounded vortex patches and the case of vertically bounded vortex layers. We develop the contour dynamics equation for the boundary of the patch in this horizontally periodic setting and demonstrate global [math] regularity of this patch boundary. In the process of formulating the problem, we consider different notions of periodic solutions of the two-dimensional incompressible Euler equations and demonstrate equivalence of these.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2286-2311, April 2024. <br/> Abstract. We study vortex patches for the two-dimensional incompressible Euler equations. Prior works on this problem take the support of the vorticity (i.e., the vortex patch) to be a bounded region. We instead consider the horizontally periodic setting. This includes both the case of a periodic array of bounded vortex patches and the case of vertically bounded vortex layers. We develop the contour dynamics equation for the boundary of the patch in this horizontally periodic setting and demonstrate global [math] regularity of this patch boundary. In the process of formulating the problem, we consider different notions of periodic solutions of the two-dimensional incompressible Euler equations and demonstrate equivalence of these.
Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers
10.1137/22M1525818
SIAM Journal on Mathematical Analysis
2024-03-14T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
David M. Ambrose
Fazel Hadadifard
James P. Kelliher
Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers
56
2
2286
2311
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1525818
https://epubs.siam.org/doi/abs/10.1137/22M1525818?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Weak and Renormalized Solutions to a Hypoelliptic Mean Field Games System
https://epubs.siam.org/doi/abs/10.1137/22M1497936?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2312-2356, April 2024. <br/> Abstract. We study the well-posedness of a degenerate, hypoelliptic Mean Field Games system with local coupling and Hamiltonians which either are Lipschitz or grow quadratically in the gradient. In the former case, we prove the existence and uniqueness of weak solutions, while in the latter we study the same question for renormalized solutions. Our approach relies on the kinetic regularity of hypoelliptic equations obtained by Bouchut and the work of Porretta on the existence and uniqueness of renormalized solutions for the nondegenerate Mean Field Games system.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2312-2356, April 2024. <br/> Abstract. We study the well-posedness of a degenerate, hypoelliptic Mean Field Games system with local coupling and Hamiltonians which either are Lipschitz or grow quadratically in the gradient. In the former case, we prove the existence and uniqueness of weak solutions, while in the latter we study the same question for renormalized solutions. Our approach relies on the kinetic regularity of hypoelliptic equations obtained by Bouchut and the work of Porretta on the existence and uniqueness of renormalized solutions for the nondegenerate Mean Field Games system.
Weak and Renormalized Solutions to a Hypoelliptic Mean Field Games System
10.1137/22M1497936
SIAM Journal on Mathematical Analysis
2024-03-15T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Nikiforos Mimikos-Stamatopoulos
Weak and Renormalized Solutions to a Hypoelliptic Mean Field Games System
56
2
2312
2356
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1497936
https://epubs.siam.org/doi/abs/10.1137/22M1497936?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Core Shells and Double Bubbles in a Weighted Nonlocal Isoperimetric Problem
https://epubs.siam.org/doi/abs/10.1137/22M1538545?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2357-2394, April 2024. <br/> Abstract. We consider a sharp-interface model of [math] triblock copolymers, for which the surface tension [math] across the interface separating phase [math] from phase [math] may depend on the components. We study global minimizers of the associated ternary local isoperimetric problem in [math], and show how the geometry of minimizers changes with the surface tensions [math], varying from symmetric double-bubbles for equal surface tensions, through asymmetric double bubbles, to core shells as the values of [math] become more disparate. Then we consider the effect of nonlocal interactions in a droplet scaling regime, in which vanishingly small particles of two phases are distributed in a sea of the third phase. We are particularly interested in a degenerate case of [math] in which minimizers exhibit core shell geometry, as this phase configuration is expected on physical grounds in nonlocal ternary systems.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2357-2394, April 2024. <br/> Abstract. We consider a sharp-interface model of [math] triblock copolymers, for which the surface tension [math] across the interface separating phase [math] from phase [math] may depend on the components. We study global minimizers of the associated ternary local isoperimetric problem in [math], and show how the geometry of minimizers changes with the surface tensions [math], varying from symmetric double-bubbles for equal surface tensions, through asymmetric double bubbles, to core shells as the values of [math] become more disparate. Then we consider the effect of nonlocal interactions in a droplet scaling regime, in which vanishingly small particles of two phases are distributed in a sea of the third phase. We are particularly interested in a degenerate case of [math] in which minimizers exhibit core shell geometry, as this phase configuration is expected on physical grounds in nonlocal ternary systems.
Core Shells and Double Bubbles in a Weighted Nonlocal Isoperimetric Problem
10.1137/22M1538545
SIAM Journal on Mathematical Analysis
2024-03-15T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Stanley Alama
Lia Bronsard
Xinyang Lu
Chong Wang
Core Shells and Double Bubbles in a Weighted Nonlocal Isoperimetric Problem
56
2
2357
2394
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1538545
https://epubs.siam.org/doi/abs/10.1137/22M1538545?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Stochastic Homogenization and Geometric Singularities: A Study on Corners
https://epubs.siam.org/doi/abs/10.1137/23M1559361?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2395-2455, April 2024. <br/> Abstract. In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in two dimensions: Unlike in the whole-space, on such a sector there exist nonsmooth harmonic functions (these depend on the angle of the sector). Here we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a logarithmic loss. Our construction of the corner correctors relies on a large-scale regularity theory for [math]-harmonic functions in the sector, which we also prove and which, as a by-product, yields a Liouville principle. We also propose a nonstandard 2-scale expansion, which is adapted to the sectoral domain and incorporates the corner correctors. Our final result is a quasi-optimal error estimate for this adapted 2-scale expansion.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2395-2455, April 2024. <br/> Abstract. In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in two dimensions: Unlike in the whole-space, on such a sector there exist nonsmooth harmonic functions (these depend on the angle of the sector). Here we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a logarithmic loss. Our construction of the corner correctors relies on a large-scale regularity theory for [math]-harmonic functions in the sector, which we also prove and which, as a by-product, yields a Liouville principle. We also propose a nonstandard 2-scale expansion, which is adapted to the sectoral domain and incorporates the corner correctors. Our final result is a quasi-optimal error estimate for this adapted 2-scale expansion.
Stochastic Homogenization and Geometric Singularities: A Study on Corners
10.1137/23M1559361
SIAM Journal on Mathematical Analysis
2024-03-18T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Marc Josien
Claudia Raithel
Mathias Schäffner
Stochastic Homogenization and Geometric Singularities: A Study on Corners
56
2
2395
2455
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1559361
https://epubs.siam.org/doi/abs/10.1137/23M1559361?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Stability Estimates for the Inverse Fractional Conductivity Problem
https://epubs.siam.org/doi/abs/10.1137/22M1533542?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2456-2487, April 2024. <br/> Abstract. We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini’s work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2456-2487, April 2024. <br/> Abstract. We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini’s work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.
Stability Estimates for the Inverse Fractional Conductivity Problem
10.1137/22M1533542
SIAM Journal on Mathematical Analysis
2024-03-19T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Giovanni Covi
Jesse Railo
Teemu Tyni
Philipp Zimmermann
Stability Estimates for the Inverse Fractional Conductivity Problem
56
2
2456
2487
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1533542
https://epubs.siam.org/doi/abs/10.1137/22M1533542?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Analytic Regularity of Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions in Polygons
https://epubs.siam.org/doi/abs/10.1137/22M1527428?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2488-2520, April 2024. <br/> Abstract. We prove weighted analytic regularity of Leray–Hopf variational solutions for the stationary, incompressible Navier–Stokes equations (NSEs) in plane polygons, subject to analytic body forces. We admit mixed boundary conditions which may change type at each corner. The weighted analytic regularity results are established in scales of corner-weighted Kondrat’ev spaces of finite order. The proofs rely on a priori estimates for the corresponding linearized boundary value problem in sectors in corner-weighted Sobolev spaces and on an induction argument for the weighted norm estimates on the quadratic nonlinear term in the NSE in a polar frame.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2488-2520, April 2024. <br/> Abstract. We prove weighted analytic regularity of Leray–Hopf variational solutions for the stationary, incompressible Navier–Stokes equations (NSEs) in plane polygons, subject to analytic body forces. We admit mixed boundary conditions which may change type at each corner. The weighted analytic regularity results are established in scales of corner-weighted Kondrat’ev spaces of finite order. The proofs rely on a priori estimates for the corresponding linearized boundary value problem in sectors in corner-weighted Sobolev spaces and on an induction argument for the weighted norm estimates on the quadratic nonlinear term in the NSE in a polar frame.
Analytic Regularity of Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions in Polygons
10.1137/22M1527428
SIAM Journal on Mathematical Analysis
2024-03-20T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Yanchen He
Carlo Marcati
Christoph Schwab
Analytic Regularity of Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions in Polygons
56
2
2488
2520
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1527428
https://epubs.siam.org/doi/abs/10.1137/22M1527428?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Stability of Black Solitons in Optical Systems with Intensity-Dependent Dispersion
https://epubs.siam.org/doi/abs/10.1137/23M1552838?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2521-2568, April 2024. <br/>Abstract. Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to the addition of a small decaying potential and remains spectrally stable when it is pinned to the minimum points of the effective potential. The same model exhibits a family of traveling dark solitons for every wave speed and we incorporate properties of these dark solitons for small wave speeds in the analysis of orbital stability of the black soliton.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2521-2568, April 2024. <br/>Abstract. Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to the addition of a small decaying potential and remains spectrally stable when it is pinned to the minimum points of the effective potential. The same model exhibits a family of traveling dark solitons for every wave speed and we incorporate properties of these dark solitons for small wave speeds in the analysis of orbital stability of the black soliton.
Stability of Black Solitons in Optical Systems with Intensity-Dependent Dispersion
10.1137/23M1552838
SIAM Journal on Mathematical Analysis
2024-03-22T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Dmitry E. Pelinovsky
Michael Plum
Stability of Black Solitons in Optical Systems with Intensity-Dependent Dispersion
56
2
2521
2568
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1552838
https://epubs.siam.org/doi/abs/10.1137/23M1552838?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Mean Field Games Master Equations: From Discrete to Continuous State Space
https://epubs.siam.org/doi/abs/10.1137/23M1552528?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2569-2610, April 2024. <br/> Abstract. This paper studies the convergence of mean field games (MFGs) with finite state space to MFGs with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochastic control, but also of finite difference numerical schemes; time remains continuous in the discretization, and the time horizon is arbitrarily long. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both when there is a smooth solution to the limit master equation and when there is not. The second approach relies on the notion of monotone solutions introduced by [, ]. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, and by compactness arguments otherwise.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2569-2610, April 2024. <br/> Abstract. This paper studies the convergence of mean field games (MFGs) with finite state space to MFGs with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochastic control, but also of finite difference numerical schemes; time remains continuous in the discretization, and the time horizon is arbitrarily long. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both when there is a smooth solution to the limit master equation and when there is not. The second approach relies on the notion of monotone solutions introduced by [, ]. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, and by compactness arguments otherwise.
Mean Field Games Master Equations: From Discrete to Continuous State Space
10.1137/23M1552528
SIAM Journal on Mathematical Analysis
2024-03-22T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Charles Bertucci
Alekos Cecchin
Mean Field Games Master Equations: From Discrete to Continuous State Space
56
2
2569
2610
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1552528
https://epubs.siam.org/doi/abs/10.1137/23M1552528?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Upper Bound for the Grand Canonical Free Energy of the Bose Gas in the Gross–Pitaevskii Limit
https://epubs.siam.org/doi/abs/10.1137/23M1580930?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2611-2660, April 2024. <br/> Abstract. We consider a homogeneous Bose gas in the Gross–Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose–Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) The free energy of the interacting condensate is given in terms of an effective theory describing its particle number fluctuations, and (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2611-2660, April 2024. <br/> Abstract. We consider a homogeneous Bose gas in the Gross–Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose–Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) The free energy of the interacting condensate is given in terms of an effective theory describing its particle number fluctuations, and (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian.
Upper Bound for the Grand Canonical Free Energy of the Bose Gas in the Gross–Pitaevskii Limit
10.1137/23M1580930
SIAM Journal on Mathematical Analysis
2024-03-26T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chiara Boccato
Andreas Deuchert
David Stocker
Upper Bound for the Grand Canonical Free Energy of the Bose Gas in the Gross–Pitaevskii Limit
56
2
2611
2660
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1580930
https://epubs.siam.org/doi/abs/10.1137/23M1580930?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Strong Convergence of Propagation of Chaos for McKean–Vlasov SDEs with Singular Interactions
https://epubs.siam.org/doi/abs/10.1137/23M1556666?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2661-2713, April 2024. <br/> Abstract. In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular [math]-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Röckner [Probab. Theory Related Fields, 131 (2005), pp. 154–196] to the case of mixed [math]-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang’s entropy method [P.-E. Jabin and Z. Wang, J. Funct. Anal., 271 (2016), pp. 3588–3627] and Zvonkin’s transformation.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2661-2713, April 2024. <br/> Abstract. In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular [math]-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Röckner [Probab. Theory Related Fields, 131 (2005), pp. 154–196] to the case of mixed [math]-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang’s entropy method [P.-E. Jabin and Z. Wang, J. Funct. Anal., 271 (2016), pp. 3588–3627] and Zvonkin’s transformation.
Strong Convergence of Propagation of Chaos for McKean–Vlasov SDEs with Singular Interactions
10.1137/23M1556666
SIAM Journal on Mathematical Analysis
2024-03-26T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Zimo Hao
Michael Röckner
Xicheng Zhang
Strong Convergence of Propagation of Chaos for McKean–Vlasov SDEs with Singular Interactions
56
2
2661
2713
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/23M1556666
https://epubs.siam.org/doi/abs/10.1137/23M1556666?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Uniqueness and Orbital Stability of Standing Waves for the Nonlinear Schrödinger Equation with a Partial Confinement
https://epubs.siam.org/doi/abs/10.1137/22M1534705?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2714-2737, April 2024. <br/> Abstract. We consider the 3d cubic nonlinear Schrödinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower-dimensional dynamics of the Bose–Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini et al. [Comm. Math. Phys. 353 (2017), pp. 229–251], a proper ground state is constructed, introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that, as the partial confinement is increased, the 1d ground state is derived from the 3d energy minimizer with a precise rate of convergence. Then, by employing this dimension reduction limit, we prove the uniqueness of the 3d minimizer, provided that the confinement is sufficiently strong. Consequently, we obtain the orbital stability of the minimizer, which improves that of the set of minimizers in the previous work Bellazini et al. [Comm. Math. Phys., 353 (2017), pp. 229–251].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2714-2737, April 2024. <br/> Abstract. We consider the 3d cubic nonlinear Schrödinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower-dimensional dynamics of the Bose–Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini et al. [Comm. Math. Phys. 353 (2017), pp. 229–251], a proper ground state is constructed, introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that, as the partial confinement is increased, the 1d ground state is derived from the 3d energy minimizer with a precise rate of convergence. Then, by employing this dimension reduction limit, we prove the uniqueness of the 3d minimizer, provided that the confinement is sufficiently strong. Consequently, we obtain the orbital stability of the minimizer, which improves that of the set of minimizers in the previous work Bellazini et al. [Comm. Math. Phys., 353 (2017), pp. 229–251].
Uniqueness and Orbital Stability of Standing Waves for the Nonlinear Schrödinger Equation with a Partial Confinement
10.1137/22M1534705
SIAM Journal on Mathematical Analysis
2024-03-28T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Younghun Hong
Sangdon Jin
Uniqueness and Orbital Stability of Standing Waves for the Nonlinear Schrödinger Equation with a Partial Confinement
56
2
2714
2737
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1534705
https://epubs.siam.org/doi/abs/10.1137/22M1534705?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Linear Elliptic Homogenization for a Class of Highly Oscillating Nonperiodic Potentials
https://epubs.siam.org/doi/abs/10.1137/22M1504275?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/2">Volume 56, Issue 2</a>, Page 2738-2782, April 2024. <br/> Abstract. We consider a homogenization problem for the second-order elliptic equation [math] when the highly oscillatory potential [math] belongs to a particular class of nonperiodic potentials. We show the existence of an adapted corrector and prove the convergence of [math] to its homogenized limit.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 2, Page 2738-2782, April 2024. <br/> Abstract. We consider a homogenization problem for the second-order elliptic equation [math] when the highly oscillatory potential [math] belongs to a particular class of nonperiodic potentials. We show the existence of an adapted corrector and prove the convergence of [math] to its homogenized limit.
Linear Elliptic Homogenization for a Class of Highly Oscillating Nonperiodic Potentials
10.1137/22M1504275
SIAM Journal on Mathematical Analysis
2024-03-28T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Rémi Goudey
Claude Le Bris
Linear Elliptic Homogenization for a Class of Highly Oscillating Nonperiodic Potentials
56
2
2738
2782
2024-04-30T07:00:00Z
2024-04-30T07:00:00Z
10.1137/22M1504275
https://epubs.siam.org/doi/abs/10.1137/22M1504275?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Large Deviations for Stochastic Generalized Porous Media Equations Driven by Lévy Noise
https://epubs.siam.org/doi/abs/10.1137/22M1506900?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1-42, February 2024. <br/> Abstract. We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a [math]-finite measure space [math], with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient [math] is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open [math], [math] Laplacian or fractional Laplacians (i.e., [math], [math]), and generalized Schrödinger operators (i.e., [math]); Laplacians on fractals is also included.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1-42, February 2024. <br/> Abstract. We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a [math]-finite measure space [math], with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient [math] is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open [math], [math] Laplacian or fractional Laplacians (i.e., [math], [math]), and generalized Schrödinger operators (i.e., [math]); Laplacians on fractals is also included.
Large Deviations for Stochastic Generalized Porous Media Equations Driven by Lévy Noise
10.1137/22M1506900
SIAM Journal on Mathematical Analysis
2024-01-03T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Weina Wu
Jianliang Zhai
Large Deviations for Stochastic Generalized Porous Media Equations Driven by Lévy Noise
56
1
1
42
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1506900
https://epubs.siam.org/doi/abs/10.1137/22M1506900?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Regularity and Long Time Behavior of One-Dimensional First-Order Mean Field Games and the Planning Problem
https://epubs.siam.org/doi/abs/10.1137/23M1547779?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 43-78, February 2024. <br/> Abstract. We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a nonseparated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume either blow-up of the cost function or that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence and identifying their limit as the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 43-78, February 2024. <br/> Abstract. We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a nonseparated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume either blow-up of the cost function or that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence and identifying their limit as the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density.
Regularity and Long Time Behavior of One-Dimensional First-Order Mean Field Games and the Planning Problem
10.1137/23M1547779
SIAM Journal on Mathematical Analysis
2024-01-03T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Nikiforos Mimikos-Stamatopoulos
Sebastian Munoz
Regularity and Long Time Behavior of One-Dimensional First-Order Mean Field Games and the Planning Problem
56
1
43
78
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1547779
https://epubs.siam.org/doi/abs/10.1137/23M1547779?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Semidiscrete Modeling of Systems of Wedge Disclinations and Edge Dislocations via the Airy Stress Function Method
https://epubs.siam.org/doi/abs/10.1137/22M1523443?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 79-136, February 2024. <br/> Abstract. We present a variational theory for lattice defects of rotational and translational type. We focus on finite systems of planar wedge disclinations, disclination dipoles, and edge dislocations, which we model as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Operating under the assumption of planar linearized kinematics, we formulate the mechanical equilibrium problem in terms of the Airy stress function, for which we introduce a rigorous analytical formulation in the context of incompatible elasticity. Our main result entails the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. By adopting the regularization approach via core radius, we show that, as the core radius vanishes, the asymptotic energy expansion for disclination dipoles coincides with the energy of finite systems of edge dislocations. This proves that Eshelby’s kinematic characterization of an edge dislocation in terms of a disclination dipole is exact also from the energetic standpoint.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 79-136, February 2024. <br/> Abstract. We present a variational theory for lattice defects of rotational and translational type. We focus on finite systems of planar wedge disclinations, disclination dipoles, and edge dislocations, which we model as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Operating under the assumption of planar linearized kinematics, we formulate the mechanical equilibrium problem in terms of the Airy stress function, for which we introduce a rigorous analytical formulation in the context of incompatible elasticity. Our main result entails the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. By adopting the regularization approach via core radius, we show that, as the core radius vanishes, the asymptotic energy expansion for disclination dipoles coincides with the energy of finite systems of edge dislocations. This proves that Eshelby’s kinematic characterization of an edge dislocation in terms of a disclination dipole is exact also from the energetic standpoint.
Semidiscrete Modeling of Systems of Wedge Disclinations and Edge Dislocations via the Airy Stress Function Method
10.1137/22M1523443
SIAM Journal on Mathematical Analysis
2024-01-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Pierluigi Cesana
Lucia De Luca
Marco Morandotti
Semidiscrete Modeling of Systems of Wedge Disclinations and Edge Dislocations via the Airy Stress Function Method
56
1
79
136
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1523443
https://epubs.siam.org/doi/abs/10.1137/22M1523443?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Asymptotics of the Hard Edge Pearcey Determinant
https://epubs.siam.org/doi/abs/10.1137/22M1513897?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 137-172, February 2024. <br/> Abstract. We study the Fredholm determinant of an integral operator associated to the hard edge Pearcey kernel. This determinant appears in a variety of random matrix and nonintersecting paths models. By relating the logarithmic derivatives of the Fredholm determinant to a [math] Riemann–Hilbert problem, we obtain asymptotics of the determinant, which is also known as the large gap asymptotics for the corresponding point process.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 137-172, February 2024. <br/> Abstract. We study the Fredholm determinant of an integral operator associated to the hard edge Pearcey kernel. This determinant appears in a variety of random matrix and nonintersecting paths models. By relating the logarithmic derivatives of the Fredholm determinant to a [math] Riemann–Hilbert problem, we obtain asymptotics of the determinant, which is also known as the large gap asymptotics for the corresponding point process.
Asymptotics of the Hard Edge Pearcey Determinant
10.1137/22M1513897
SIAM Journal on Mathematical Analysis
2024-01-04T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Luming Yao
Lun Zhang
Asymptotics of the Hard Edge Pearcey Determinant
56
1
137
172
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1513897
https://epubs.siam.org/doi/abs/10.1137/22M1513897?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Interface Propagation Properties for a Nonlocal Thin-Film Equation
https://epubs.siam.org/doi/abs/10.1137/22M1510297?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 173-196, February 2024. <br/> Abstract. We consider a degenerate nonlocal parabolic equation in a one-dimensional domain introduced to model hydraulic fractures. The nonlocal operator is given by a fractional power of the Laplacian and the degenerate mobility exponent corresponds to a “strong slippage” regime with “complete wetting” interfacial conditions for local thin-film equations. Using a localized entropy estimate and a Stampacchia-type lemma, we establish a finite speed of propagation result and sufficient conditions (and lower bounds) for the waiting-time phenomenon.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 173-196, February 2024. <br/> Abstract. We consider a degenerate nonlocal parabolic equation in a one-dimensional domain introduced to model hydraulic fractures. The nonlocal operator is given by a fractional power of the Laplacian and the degenerate mobility exponent corresponds to a “strong slippage” regime with “complete wetting” interfacial conditions for local thin-film equations. Using a localized entropy estimate and a Stampacchia-type lemma, we establish a finite speed of propagation result and sufficient conditions (and lower bounds) for the waiting-time phenomenon.
Interface Propagation Properties for a Nonlocal Thin-Film Equation
10.1137/22M1510297
SIAM Journal on Mathematical Analysis
2024-01-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Nicola De Nitti
Roman M. Taranets
Interface Propagation Properties for a Nonlocal Thin-Film Equation
56
1
173
196
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1510297
https://epubs.siam.org/doi/abs/10.1137/22M1510297?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Discrete Dislocation Dynamics with Annihilation as the Limit of the Peierls–Nabarro Model in One Dimension
https://epubs.siam.org/doi/abs/10.1137/22M1527052?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 197-233, February 2024. <br/> Abstract. Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls–Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls–Nabarro model which we consider describes the evolution of a phase-field function [math] which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen–Cahn type with the half-Laplacian. The small parameter [math] is the ratio between the atomic distance and the typical distance between phase transitions in [math]. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function [math] as [math] up to arbitrary end time [math]; in particular, beyond collisions. We prove that [math] converges to a piecewise constant function [math], whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and super-solutions of [math], and to show that they converge to the same limit [math].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 197-233, February 2024. <br/> Abstract. Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls–Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls–Nabarro model which we consider describes the evolution of a phase-field function [math] which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen–Cahn type with the half-Laplacian. The small parameter [math] is the ratio between the atomic distance and the typical distance between phase transitions in [math]. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function [math] as [math] up to arbitrary end time [math]; in particular, beyond collisions. We prove that [math] converges to a piecewise constant function [math], whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and super-solutions of [math], and to show that they converge to the same limit [math].
Discrete Dislocation Dynamics with Annihilation as the Limit of the Peierls–Nabarro Model in One Dimension
10.1137/22M1527052
SIAM Journal on Mathematical Analysis
2024-01-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Patrick van Meurs
Stefania Patrizi
Discrete Dislocation Dynamics with Annihilation as the Limit of the Peierls–Nabarro Model in One Dimension
56
1
197
233
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1527052
https://epubs.siam.org/doi/abs/10.1137/22M1527052?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations
https://epubs.siam.org/doi/abs/10.1137/23M1563360?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 234-253, February 2024. <br/> Abstract. In this paper, we are interested in the following degenerate elliptic Monge–Ampère equation:[math] Under suitable structure conditions on [math], we can show that [math] and the solutions of linearized equation have the same symmetric property as the domain. Moreover, we also achieve some uniqueness results for homogenous nonlinearity [math] ([math]) in general convex domain. Compared to the results in [G. Huang, Calc. Var. Partial Differential Equations, 58 (2019), 73], we dispose the “uniformly convex” condition imposed on the domain. With a subtle approximation procedure and variant form of Hopf’s lemma, we overcome the technical difficulties caused by the degeneracy of the above equation. What’s more, we can also get the uniqueness property for domains whose geometry is close enough to the symmetric convex domain.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 234-253, February 2024. <br/> Abstract. In this paper, we are interested in the following degenerate elliptic Monge–Ampère equation:[math] Under suitable structure conditions on [math], we can show that [math] and the solutions of linearized equation have the same symmetric property as the domain. Moreover, we also achieve some uniqueness results for homogenous nonlinearity [math] ([math]) in general convex domain. Compared to the results in [G. Huang, Calc. Var. Partial Differential Equations, 58 (2019), 73], we dispose the “uniformly convex” condition imposed on the domain. With a subtle approximation procedure and variant form of Hopf’s lemma, we overcome the technical difficulties caused by the degeneracy of the above equation. What’s more, we can also get the uniqueness property for domains whose geometry is close enough to the symmetric convex domain.
Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations
10.1137/23M1563360
SIAM Journal on Mathematical Analysis
2024-01-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Tingzhi Cheng
Genggeng Huang
Xianghui Xu
Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations
56
1
234
253
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1563360
https://epubs.siam.org/doi/abs/10.1137/23M1563360?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise
https://epubs.siam.org/doi/abs/10.1137/22M1529129?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 254-274, February 2024. <br/> Abstract. The aim of this paper is to establish the asymptotically autonomous robustness of pullback random attractors of nonautonomous Benjamin–Bona–Mahony equations driven by nonlinear colored noise defined on 3D unbounded channels. We first prove the existence, uniqueness, and backward compactness of a special kind of pullback random attractor by the methods of spectral decomposition inside bounded domains as well as the uniform tail-estimates of solutions outside bounded domains over the infinite time interval in order to surmount the difficulties caused by lack of compact Sobolev embedding on unbounded domains and weak dissipative structure of the equation. The measurability of such an attractor is proven by showing that the defined two kinds of attractors with respect to two different universes are equal. Finally, the asymptotically autonomous upper semicontinuity of the attractors is investigated by assuming that the time-dependent external forcing term converges to the time-independent external force as the time-parameter tends to negative infinity. This work is a continuation of our previous work [Chen et al., Math. Ann., 386 (2023), pp. 343–373], which considered the existence and uniqueness of usual tempered pullback random attractors.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 254-274, February 2024. <br/> Abstract. The aim of this paper is to establish the asymptotically autonomous robustness of pullback random attractors of nonautonomous Benjamin–Bona–Mahony equations driven by nonlinear colored noise defined on 3D unbounded channels. We first prove the existence, uniqueness, and backward compactness of a special kind of pullback random attractor by the methods of spectral decomposition inside bounded domains as well as the uniform tail-estimates of solutions outside bounded domains over the infinite time interval in order to surmount the difficulties caused by lack of compact Sobolev embedding on unbounded domains and weak dissipative structure of the equation. The measurability of such an attractor is proven by showing that the defined two kinds of attractors with respect to two different universes are equal. Finally, the asymptotically autonomous upper semicontinuity of the attractors is investigated by assuming that the time-dependent external forcing term converges to the time-independent external force as the time-parameter tends to negative infinity. This work is a continuation of our previous work [Chen et al., Math. Ann., 386 (2023), pp. 343–373], which considered the existence and uniqueness of usual tempered pullback random attractors.
Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise
10.1137/22M1529129
SIAM Journal on Mathematical Analysis
2024-01-05T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Pengyu Chen
Renhai Wang
Xuping Zhang
Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise
56
1
254
274
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1529129
https://epubs.siam.org/doi/abs/10.1137/22M1529129?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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The p–Laplace “Signature” for Quasilinear Inverse Problems with Large Boundary Data
https://epubs.siam.org/doi/abs/10.1137/22M1529154?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 275-303, February 2024. <br/> Abstract. This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. We consider nonlinear constitutive relationships which, at a given point in the space, present a behavior for large arguments that is described by monomials of order [math] and [math]. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of this work is that for “large” Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted [math]Laplace problem.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 275-303, February 2024. <br/> Abstract. This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. We consider nonlinear constitutive relationships which, at a given point in the space, present a behavior for large arguments that is described by monomials of order [math] and [math]. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of this work is that for “large” Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted [math]Laplace problem.
The p–Laplace “Signature” for Quasilinear Inverse Problems with Large Boundary Data
10.1137/22M1529154
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Antonio Corbo Esposito
Luisa Faella
Gianpaolo Piscitelli
Ravi Prakash
Antonello Tamburrino
The p–Laplace “Signature” for Quasilinear Inverse Problems with Large Boundary Data
56
1
275
303
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1529154
https://epubs.siam.org/doi/abs/10.1137/22M1529154?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Dissipative Measure-Valued Solutions to the Euler–Poisson Equation
https://epubs.siam.org/doi/abs/10.1137/22M1525983?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 304-335, February 2024. <br/> Abstract. We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsion-attraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 304-335, February 2024. <br/> Abstract. We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsion-attraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.
Dissipative Measure-Valued Solutions to the Euler–Poisson Equation
10.1137/22M1525983
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
José A. Carrillo
Tomasz Dębiec
Piotr Gwiazda
Agnieszka Świerczewska-Gwiazda
Dissipative Measure-Valued Solutions to the Euler–Poisson Equation
56
1
304
335
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1525983
https://epubs.siam.org/doi/abs/10.1137/22M1525983?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators
https://epubs.siam.org/doi/abs/10.1137/22M1511503?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 336-373, February 2024. <br/> Abstract. We consider a class of elliptic and parabolic problems featuring a specific nonlocal operator of fractional Laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 336-373, February 2024. <br/> Abstract. We consider a class of elliptic and parabolic problems featuring a specific nonlocal operator of fractional Laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.
Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators
10.1137/22M1511503
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Stefano Buccheri
Ulisse Stefanelli
Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators
56
1
336
373
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1511503
https://epubs.siam.org/doi/abs/10.1137/22M1511503?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Generalized Beale–Kato–Majda Breakdown Criterion for the Free-Boundary Problem in Euler Equations with Surface Tension
https://epubs.siam.org/doi/abs/10.1137/23M1563761?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 374-411, February 2024. <br/> Abstract. It is shown in Ferrari [Comm. Math. Phys., 155 (1993), pp. 277–294] that if [math] is the maximal time interval of existence of a smooth solution of the incompressible Euler equations in a bounded, simply connected domain in [math], then [math], where [math] is the vorticity of the flow. Ferrari’s result generalizes the classical Beale–Kato–Majda [Comm. Math. Phys., 94 (1984), pp. 61–66] breakdown criterion in the case of a bounded fluid domain. In this manuscript, we show a breakdown criterion for a smooth solution of the Euler equations describing the motion of an incompressible fluid in a bounded domain in [math] with a free surface boundary. The fluid is under the influence of surface tension. In addition, we show that our breakdown criterion reduces to the one proved by Ferrari [Comm. Math. Phys., 155 (1993), pp. 277–294] when the free surface boundary is fixed. Specifically, the additional control norms on the moving boundary will either become trivial or stop showing up if the kinematic boundary condition on the moving boundary reduces to the slip boundary condition.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 374-411, February 2024. <br/> Abstract. It is shown in Ferrari [Comm. Math. Phys., 155 (1993), pp. 277–294] that if [math] is the maximal time interval of existence of a smooth solution of the incompressible Euler equations in a bounded, simply connected domain in [math], then [math], where [math] is the vorticity of the flow. Ferrari’s result generalizes the classical Beale–Kato–Majda [Comm. Math. Phys., 94 (1984), pp. 61–66] breakdown criterion in the case of a bounded fluid domain. In this manuscript, we show a breakdown criterion for a smooth solution of the Euler equations describing the motion of an incompressible fluid in a bounded domain in [math] with a free surface boundary. The fluid is under the influence of surface tension. In addition, we show that our breakdown criterion reduces to the one proved by Ferrari [Comm. Math. Phys., 155 (1993), pp. 277–294] when the free surface boundary is fixed. Specifically, the additional control norms on the moving boundary will either become trivial or stop showing up if the kinematic boundary condition on the moving boundary reduces to the slip boundary condition.
A Generalized Beale–Kato–Majda Breakdown Criterion for the Free-Boundary Problem in Euler Equations with Surface Tension
10.1137/23M1563761
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chenyun Luo
Kai Zhou
A Generalized Beale–Kato–Majda Breakdown Criterion for the Free-Boundary Problem in Euler Equations with Surface Tension
56
1
374
411
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1563761
https://epubs.siam.org/doi/abs/10.1137/23M1563761?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Forced Rapidly Dissipative Navier–Stokes Flows
https://epubs.siam.org/doi/abs/10.1137/22M1536807?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 412-432, February 2024. <br/> Abstract. We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier–Stokes equations in [math]. The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 412-432, February 2024. <br/> Abstract. We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier–Stokes equations in [math]. The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.
Forced Rapidly Dissipative Navier–Stokes Flows
10.1137/22M1536807
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Lorenzo Brandolese
Takahiro Okabe
Forced Rapidly Dissipative Navier–Stokes Flows
56
1
412
432
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1536807
https://epubs.siam.org/doi/abs/10.1137/22M1536807?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Stability for Time-Domain Elastic Wave Equations
https://epubs.siam.org/doi/abs/10.1137/22M1508546?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 433-453, February 2024. <br/> Abstract. This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded [math]-dimensional domain. First, an explicit formula for the density reconstruction is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 433-453, February 2024. <br/> Abstract. This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded [math]-dimensional domain. First, an explicit formula for the density reconstruction is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.
Stability for Time-Domain Elastic Wave Equations
10.1137/22M1508546
SIAM Journal on Mathematical Analysis
2024-01-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Bochao Chen
Yixian Gao
Shuguan Ji
Yang Liu
Stability for Time-Domain Elastic Wave Equations
56
1
433
453
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1508546
https://epubs.siam.org/doi/abs/10.1137/22M1508546?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Polarized High-frequency Wave Propagation Beyond the Nonlinear Schrödinger Approximation
https://epubs.siam.org/doi/abs/10.1137/22M1504810?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 454-473, February 2024. <br/> Abstract. This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein–Gordon equations and the Maxwell–Lorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear Schrödinger equation. The given construction of polarized solutions is explicit, uses in addition a linear Schrödinger equation for each further order of approximation, and is accessible to direct numerical approximation.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 454-473, February 2024. <br/> Abstract. This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein–Gordon equations and the Maxwell–Lorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear Schrödinger equation. The given construction of polarized solutions is explicit, uses in addition a linear Schrödinger equation for each further order of approximation, and is accessible to direct numerical approximation.
Polarized High-frequency Wave Propagation Beyond the Nonlinear Schrödinger Approximation
10.1137/22M1504810
SIAM Journal on Mathematical Analysis
2024-01-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Julian Baumstark
Tobias Jahnke
Christian Lubich
Polarized High-frequency Wave Propagation Beyond the Nonlinear Schrödinger Approximation
56
1
454
473
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1504810
https://epubs.siam.org/doi/abs/10.1137/22M1504810?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Sharp Behavior of Dirichlet–Laplacian Eigenvalues for a Class of Singularly Perturbed Problems
https://epubs.siam.org/doi/abs/10.1137/23M1564444?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 474-500, February 2024. <br/> Abstract. We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation’s magnitude. It is a sort of torsional rigidity of the tube’s section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a proper way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 474-500, February 2024. <br/> Abstract. We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation’s magnitude. It is a sort of torsional rigidity of the tube’s section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a proper way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.
Sharp Behavior of Dirichlet–Laplacian Eigenvalues for a Class of Singularly Perturbed Problems
10.1137/23M1564444
SIAM Journal on Mathematical Analysis
2024-01-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Laura Abatangelo
Roberto Ognibene
Sharp Behavior of Dirichlet–Laplacian Eigenvalues for a Class of Singularly Perturbed Problems
56
1
474
500
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1564444
https://epubs.siam.org/doi/abs/10.1137/23M1564444?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Global Existence for the Stochastic Boussinesq Equations with Transport Noise and Small Rough Data
https://epubs.siam.org/doi/abs/10.1137/23M1559531?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 501-528, February 2024. <br/> Abstract. In this paper, we consider the stochastic Boussinesq equations on [math] with transport noise and rough initial data. We prove the existence and uniqueness of the local pathwise solution with initial data in [math] for [math]. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 501-528, February 2024. <br/> Abstract. In this paper, we consider the stochastic Boussinesq equations on [math] with transport noise and rough initial data. We prove the existence and uniqueness of the local pathwise solution with initial data in [math] for [math]. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.
Global Existence for the Stochastic Boussinesq Equations with Transport Noise and Small Rough Data
10.1137/23M1559531
SIAM Journal on Mathematical Analysis
2024-01-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Quyuan Lin
Rongchang Liu
Weinan Wang
Global Existence for the Stochastic Boussinesq Equations with Transport Noise and Small Rough Data
56
1
501
528
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1559531
https://epubs.siam.org/doi/abs/10.1137/23M1559531?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Mean Field Games Systems under Displacement Monotonicity
https://epubs.siam.org/doi/abs/10.1137/22M1534353?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 529-553, February 2024. <br/> Abstract. In this note we prove the uniqueness of solutions to a class of mean field games systems subject to possibly degenerate individual noise. Our results hold true for arbitrary long time horizons and for general nonseparable Hamiltonians that satisfy a so-called displacement monotonicity condition. This monotonicity condition that we propose for nonseparable Hamiltonians is sharper and more general than the one proposed in the work [W. Gangbo et al., Ann. Probab., 50 (2022), pp. 2178–2217]. The displacement monotonicity assumptions imposed on the data actually provide not only uniqueness, but also the existence and regularity of the solutions. Our analysis uses elementary arguments and does not rely on the well-posedness of the corresponding master equations.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 529-553, February 2024. <br/> Abstract. In this note we prove the uniqueness of solutions to a class of mean field games systems subject to possibly degenerate individual noise. Our results hold true for arbitrary long time horizons and for general nonseparable Hamiltonians that satisfy a so-called displacement monotonicity condition. This monotonicity condition that we propose for nonseparable Hamiltonians is sharper and more general than the one proposed in the work [W. Gangbo et al., Ann. Probab., 50 (2022), pp. 2178–2217]. The displacement monotonicity assumptions imposed on the data actually provide not only uniqueness, but also the existence and regularity of the solutions. Our analysis uses elementary arguments and does not rely on the well-posedness of the corresponding master equations.
Mean Field Games Systems under Displacement Monotonicity
10.1137/22M1534353
SIAM Journal on Mathematical Analysis
2024-01-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Alpár R. Mészáros
Chenchen Mou
Mean Field Games Systems under Displacement Monotonicity
56
1
529
553
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1534353
https://epubs.siam.org/doi/abs/10.1137/22M1534353?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Improved Concentration of Laguerre and Jacobi Ensembles
https://epubs.siam.org/doi/abs/10.1137/23M1545343?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 554-567, February 2024. <br/> Abstract. We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that, up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 554-567, February 2024. <br/> Abstract. We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that, up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.
Improved Concentration of Laguerre and Jacobi Ensembles
10.1137/23M1545343
SIAM Journal on Mathematical Analysis
2024-01-09T08:00:00Z
© 2024 Yichen Huang and Aram W. Harrow
Yichen Huang (黄溢辰)
Aram W. Harrow
Improved Concentration of Laguerre and Jacobi Ensembles
56
1
554
567
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1545343
https://epubs.siam.org/doi/abs/10.1137/23M1545343?ai=s2&mi=3bfys9&af=R
© 2024 Yichen Huang and Aram W. Harrow
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Kinetic Chemotaxis Tumbling Kernel Determined from Macroscopic Quantities
https://epubs.siam.org/doi/abs/10.1137/22M1499911?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 568-587, February 2024. <br/> Abstract. Chemotaxis is the physical phenomenon that bacteria adjust their motions according to chemical stimulus. A classical model for this phenomenon is a kinetic equation that describes the velocity jump process whose tumbling/transition kernel uniquely determines the effect of a chemical stimulus on bacteria. The model has been shown to be an accurate model that matches with bacteria motion qualitatively. For a quantitative modeling, biophysicists and practitioners are also highly interested in determining the explicit value of the tumbling kernel. Due to the experimental limitations, measurements are typically macroscopic in nature. Do macroscopic quantities contain enough information to recover microscopic behavior? In this paper, we give a positive answer. We show that when given a special design of initial data, the population density, one specific macroscopic quantity as a function of time, contains sufficient information to recover the tumbling kernel and its associated damping coefficient. Moreover, we can read off the chemotaxis tumbling kernel using the values of population density directly from this specific experimental design. This theoretical result using kinetic theory sheds light on how practitioners may conduct experiments in laboratories.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 568-587, February 2024. <br/> Abstract. Chemotaxis is the physical phenomenon that bacteria adjust their motions according to chemical stimulus. A classical model for this phenomenon is a kinetic equation that describes the velocity jump process whose tumbling/transition kernel uniquely determines the effect of a chemical stimulus on bacteria. The model has been shown to be an accurate model that matches with bacteria motion qualitatively. For a quantitative modeling, biophysicists and practitioners are also highly interested in determining the explicit value of the tumbling kernel. Due to the experimental limitations, measurements are typically macroscopic in nature. Do macroscopic quantities contain enough information to recover microscopic behavior? In this paper, we give a positive answer. We show that when given a special design of initial data, the population density, one specific macroscopic quantity as a function of time, contains sufficient information to recover the tumbling kernel and its associated damping coefficient. Moreover, we can read off the chemotaxis tumbling kernel using the values of population density directly from this specific experimental design. This theoretical result using kinetic theory sheds light on how practitioners may conduct experiments in laboratories.
Kinetic Chemotaxis Tumbling Kernel Determined from Macroscopic Quantities
10.1137/22M1499911
SIAM Journal on Mathematical Analysis
2024-01-10T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Kathrin Hellmuth
Christian Klingenberg
Qin Li
Min Tang
Kinetic Chemotaxis Tumbling Kernel Determined from Macroscopic Quantities
56
1
568
587
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1499911
https://epubs.siam.org/doi/abs/10.1137/22M1499911?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Stability of the Ball for Attractive-Repulsive Energies
https://epubs.siam.org/doi/abs/10.1137/22M1506894?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 588-615, February 2024. <br/> Abstract. We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by Frank and Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in the [math]-sense.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 588-615, February 2024. <br/> Abstract. We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by Frank and Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in the [math]-sense.
Stability of the Ball for Attractive-Repulsive Energies
10.1137/22M1506894
SIAM Journal on Mathematical Analysis
2024-01-11T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Marco Bonacini
Riccardo Cristoferi
Ihsan Topaloglu
Stability of the Ball for Attractive-Repulsive Energies
56
1
588
615
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1506894
https://epubs.siam.org/doi/abs/10.1137/22M1506894?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Lipschitz Stability Estimate and Uniqueness in the Retrospective Analysis for the Mean Field Games System via Two Carleman Estimates
https://epubs.siam.org/doi/abs/10.1137/23M1554801?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 616-636, February 2024. <br/> Abstract. A retrospective analysis process for the mean field games system (MFGS) is considered. For the first time, Carleman estimates are applied to the analysis of the MFGS. Two new Carleman estimates are derived. They allow us to obtain the Lipschitz stability estimate with respect to a possible error in the input initial and terminal data of a retrospective problem for MFGS. This stability estimate, in turn, implies a uniqueness theorem for the problem under consideration. The idea of using Carleman estimates to obtain stability and uniqueness results came from the field of ill-posed and inverse problems.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 616-636, February 2024. <br/> Abstract. A retrospective analysis process for the mean field games system (MFGS) is considered. For the first time, Carleman estimates are applied to the analysis of the MFGS. Two new Carleman estimates are derived. They allow us to obtain the Lipschitz stability estimate with respect to a possible error in the input initial and terminal data of a retrospective problem for MFGS. This stability estimate, in turn, implies a uniqueness theorem for the problem under consideration. The idea of using Carleman estimates to obtain stability and uniqueness results came from the field of ill-posed and inverse problems.
Lipschitz Stability Estimate and Uniqueness in the Retrospective Analysis for the Mean Field Games System via Two Carleman Estimates
10.1137/23M1554801
SIAM Journal on Mathematical Analysis
2024-01-11T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Michael V. Klibanov
Yurii Averboukh
Lipschitz Stability Estimate and Uniqueness in the Retrospective Analysis for the Mean Field Games System via Two Carleman Estimates
56
1
616
636
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1554801
https://epubs.siam.org/doi/abs/10.1137/23M1554801?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Entropy Solutions to the Dirichlet Problem for Nonlinear Diffusion Equations with Conservative Noise
https://epubs.siam.org/doi/abs/10.1137/22M1537667?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 637-675, February 2024. <br/> Abstract. Motivated by porous medium equations with a randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness, and [math]-stability of nonnegative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov’s doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 637-675, February 2024. <br/> Abstract. Motivated by porous medium equations with a randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness, and [math]-stability of nonnegative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov’s doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations.
Entropy Solutions to the Dirichlet Problem for Nonlinear Diffusion Equations with Conservative Noise
10.1137/22M1537667
SIAM Journal on Mathematical Analysis
2024-01-11T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Kai Du
Ruoyang Liu
Yuxing Wang
Entropy Solutions to the Dirichlet Problem for Nonlinear Diffusion Equations with Conservative Noise
56
1
637
675
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1537667
https://epubs.siam.org/doi/abs/10.1137/22M1537667?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit
https://epubs.siam.org/doi/abs/10.1137/22M1482913?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 676-726, February 2024. <br/> Abstract. The voltage-conductance kinetic model for the collective behavior of neurons has been studied by scientists and mathematicians for two decades, but the rigorous analysis of its solution structure has been only partially obtained in spite of plenty of numerical evidence in various scenarios. In this work, we consider a simplified voltage-conductance model in which the velocity field in the voltage variable is in a separable form. The long time behavior of the simplified model is fully investigated, leading to the following dichotomy: either the density function converges to the global equilibrium, or the firing rate diverges as time goes to infinity. Besides, the fast conductance asymptotic limit is justified and analyzed, where the solution to the limit model either blows up in finite time or globally exists leading to time periodic solutions. An important implication of these results is that the nonseparable velocity field, or physically the leaky mechanism, is a key element for the emergence of periodic solutions in the original model.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 676-726, February 2024. <br/> Abstract. The voltage-conductance kinetic model for the collective behavior of neurons has been studied by scientists and mathematicians for two decades, but the rigorous analysis of its solution structure has been only partially obtained in spite of plenty of numerical evidence in various scenarios. In this work, we consider a simplified voltage-conductance model in which the velocity field in the voltage variable is in a separable form. The long time behavior of the simplified model is fully investigated, leading to the following dichotomy: either the density function converges to the global equilibrium, or the firing rate diverges as time goes to infinity. Besides, the fast conductance asymptotic limit is justified and analyzed, where the solution to the limit model either blows up in finite time or globally exists leading to time periodic solutions. An important implication of these results is that the nonseparable velocity field, or physically the leaky mechanism, is a key element for the emergence of periodic solutions in the original model.
A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit
10.1137/22M1482913
SIAM Journal on Mathematical Analysis
2024-01-11T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
José A. Carrillo
Xu’an Dou
Zhennan Zhou
A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit
56
1
676
726
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1482913
https://epubs.siam.org/doi/abs/10.1137/22M1482913?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Reconstruction of Cracks in Calderón’s Inverse Conductivity Problem Using Energy Comparisons
https://epubs.siam.org/doi/abs/10.1137/23M1572416?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 727-745, February 2024. <br/> Abstract. We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calderón’s inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 727-745, February 2024. <br/> Abstract. We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calderón’s inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.
Reconstruction of Cracks in Calderón’s Inverse Conductivity Problem Using Energy Comparisons
10.1137/23M1572416
SIAM Journal on Mathematical Analysis
2024-01-12T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Henrik Garde
Michael Vogelius
Reconstruction of Cracks in Calderón’s Inverse Conductivity Problem Using Energy Comparisons
56
1
727
745
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1572416
https://epubs.siam.org/doi/abs/10.1137/23M1572416?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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On a Model of an Elastic Body Fully Immersed in a Viscous Incompressible Fluid with Small Data
https://epubs.siam.org/doi/abs/10.1137/22M151947X?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 746-761, February 2024. <br/> Abstract. We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement [math] is governed by the damped wave equation [math] without any stabilization terms, where [math], and the fluid is modeled by the Navier–Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a priori estimates that provide the global-in-time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 746-761, February 2024. <br/> Abstract. We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement [math] is governed by the damped wave equation [math] without any stabilization terms, where [math], and the fluid is modeled by the Navier–Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a priori estimates that provide the global-in-time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.
On a Model of an Elastic Body Fully Immersed in a Viscous Incompressible Fluid with Small Data
10.1137/22M151947X
SIAM Journal on Mathematical Analysis
2024-01-12T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Igor Kukavica
Wojciech S. Ożański
On a Model of an Elastic Body Fully Immersed in a Viscous Incompressible Fluid with Small Data
56
1
746
761
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M151947X
https://epubs.siam.org/doi/abs/10.1137/22M151947X?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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An [math] Approach for the Non-Cutoff Boltzmann Equation in [math]
https://epubs.siam.org/doi/abs/10.1137/22M1533232?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 762-800, February 2024. <br/> Abstract. In the paper, we develop an [math] approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in [math]. In particular, only smallness of [math] with [math] is imposed on initial data [math], where [math] is the Fourier transform in space variable. This provides the first result on the global existence of such low-regularity solutions without relying on Sobolev embedding [math] in the case of the whole space. Different from the use of sufficiently smooth Sobolev spaces in the classical results [P. T. Gressman and R. M. Strain, J. Amer. Math. Soc., 24 (2011), pp. 771–847] and [R. Alexandre et al., J. Funct. Anal., 262 (2012), pp. 915–1010], there is a crucial difference between the torus case and the whole space case for low-regularity solutions under consideration. In fact, for the former, it is enough to take the only [math] norm corresponding to the Weiner space as studied in [R. J. Duan et al., Comm. Pure Appl. Math., 74 (2021), pp. 932–1020]. In contrast, for the latter, the extra interplay with the [math] norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of the [math] norm helps gain an almost optimal decay rate [math] of the [math] norm via the time-weighted energy estimates in the spirit of the idea of [S. Kawashima, J. Hyperbolic Differ. Equ., 1 (2004), pp. 581–603] and, in turn, this is necessarily used for establishing the global existence.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 762-800, February 2024. <br/> Abstract. In the paper, we develop an [math] approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in [math]. In particular, only smallness of [math] with [math] is imposed on initial data [math], where [math] is the Fourier transform in space variable. This provides the first result on the global existence of such low-regularity solutions without relying on Sobolev embedding [math] in the case of the whole space. Different from the use of sufficiently smooth Sobolev spaces in the classical results [P. T. Gressman and R. M. Strain, J. Amer. Math. Soc., 24 (2011), pp. 771–847] and [R. Alexandre et al., J. Funct. Anal., 262 (2012), pp. 915–1010], there is a crucial difference between the torus case and the whole space case for low-regularity solutions under consideration. In fact, for the former, it is enough to take the only [math] norm corresponding to the Weiner space as studied in [R. J. Duan et al., Comm. Pure Appl. Math., 74 (2021), pp. 932–1020]. In contrast, for the latter, the extra interplay with the [math] norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of the [math] norm helps gain an almost optimal decay rate [math] of the [math] norm via the time-weighted energy estimates in the spirit of the idea of [S. Kawashima, J. Hyperbolic Differ. Equ., 1 (2004), pp. 581–603] and, in turn, this is necessarily used for establishing the global existence.
An [math] Approach for the Non-Cutoff Boltzmann Equation in [math]
10.1137/22M1533232
SIAM Journal on Mathematical Analysis
2024-01-16T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Renjun Duan
Shota Sakamoto
Yoshihiro Ueda
An [math] Approach for the Non-Cutoff Boltzmann Equation in [math]
56
1
762
800
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1533232
https://epubs.siam.org/doi/abs/10.1137/22M1533232?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Weakly Nonlinear Geometric Optics for the Westervelt Equation and Recovery of the Nonlinearity
https://epubs.siam.org/doi/abs/10.1137/22M1543379?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 801-819, February 2024. <br/> Abstract. We study the nondiffusive Westervelt equation in the weakly nonlinear regime and show that the leading profile equation is of Burgers’ type. We show that a compactly supported nonlinearity coefficient [math] can be reconstructed from the tilt of the transmitted high frequency wave packets sent from different directions, by demonstrating that those tilts are proportional to the X-ray transform of [math].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 801-819, February 2024. <br/> Abstract. We study the nondiffusive Westervelt equation in the weakly nonlinear regime and show that the leading profile equation is of Burgers’ type. We show that a compactly supported nonlinearity coefficient [math] can be reconstructed from the tilt of the transmitted high frequency wave packets sent from different directions, by demonstrating that those tilts are proportional to the X-ray transform of [math].
Weakly Nonlinear Geometric Optics for the Westervelt Equation and Recovery of the Nonlinearity
10.1137/22M1543379
SIAM Journal on Mathematical Analysis
2024-01-16T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Nikolas Eptaminitakis
Plamen Stefanov
Weakly Nonlinear Geometric Optics for the Westervelt Equation and Recovery of the Nonlinearity
56
1
801
819
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1543379
https://epubs.siam.org/doi/abs/10.1137/22M1543379?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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The Vlasov–Poisson–Boltzmann/Landau System with Polynomial Perturbation Near Maxwellian
https://epubs.siam.org/doi/abs/10.1137/23M1567060?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 820-876, February 2024. <br/> Abstract. In this work, we consider the Vlasov–Poisson–Boltzmann system without angular cutoff and the Vlasov–Poisson–Landau system with Coulomb potential near a global Maxwellian [math] in a torus or union of cubes. We establish the global existence, uniqueness, and large-time behavior for solutions in a polynomial-weighted Sobolev space [math] for some constant [math]. For the domain union of cubes, We will consider the specular-reflection boundary condition and its high-order compatible specular boundary condition. The proof is based on an extra dissipation term generated from an improved semigroup method including the electrostatic field with the help of macroscopic estimates.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 820-876, February 2024. <br/> Abstract. In this work, we consider the Vlasov–Poisson–Boltzmann system without angular cutoff and the Vlasov–Poisson–Landau system with Coulomb potential near a global Maxwellian [math] in a torus or union of cubes. We establish the global existence, uniqueness, and large-time behavior for solutions in a polynomial-weighted Sobolev space [math] for some constant [math]. For the domain union of cubes, We will consider the specular-reflection boundary condition and its high-order compatible specular boundary condition. The proof is based on an extra dissipation term generated from an improved semigroup method including the electrostatic field with the help of macroscopic estimates.
The Vlasov–Poisson–Boltzmann/Landau System with Polynomial Perturbation Near Maxwellian
10.1137/23M1567060
SIAM Journal on Mathematical Analysis
2024-01-16T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chuqi Cao
Dingqun Deng
Xingyu Li
The Vlasov–Poisson–Boltzmann/Landau System with Polynomial Perturbation Near Maxwellian
56
1
820
876
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1567060
https://epubs.siam.org/doi/abs/10.1137/23M1567060?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions
https://epubs.siam.org/doi/abs/10.1137/22M1501040?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 877-912, February 2024. <br/> Abstract. In this paper, we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani–Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 877-912, February 2024. <br/> Abstract. In this paper, we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani–Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.
Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions
10.1137/22M1501040
SIAM Journal on Mathematical Analysis
2024-01-17T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Shengda Zeng
Vicenţiu D. Rădulescu
Patrick Winkert
Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions
56
1
877
912
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1501040
https://epubs.siam.org/doi/abs/10.1137/22M1501040?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Quantitative Coarse-Graining of Markov Chains
https://epubs.siam.org/doi/abs/10.1137/22M1473996?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 913-954, February 2024. <br/> Abstract. Coarse-graining techniques play a central role in reducing the complexity of stochastic models and are typically characterized by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyze an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 913-954, February 2024. <br/> Abstract. Coarse-graining techniques play a central role in reducing the complexity of stochastic models and are typically characterized by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyze an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.
Quantitative Coarse-Graining of Markov Chains
10.1137/22M1473996
SIAM Journal on Mathematical Analysis
2024-01-18T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Bastian Hilder
Upanshu Sharma
Quantitative Coarse-Graining of Markov Chains
56
1
913
954
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1473996
https://epubs.siam.org/doi/abs/10.1137/22M1473996?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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The Magnetic Liouville Equation as a Semiclassical Limit
https://epubs.siam.org/doi/abs/10.1137/22M1528562?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 955-992, February 2024. <br/> Abstract. The Liouville equation with nonconstant magnetic field is obtained as a limit in the Planck constant [math] of the von Neumann equation with the same magnetic field. The convergence is with respect to an appropriate semiclassical pseudodistance, and consequently with respect to the Monge–Kantorovich distance. Uniform estimates both in [math] and [math] are proved for the specific 2D case of a magnetic vector potential of the form [math]. As an application, an observation inequality for the von Neumann equation with a magnetic vector potential is obtained. These results are a magnetic variant of the works [F. Golse and T. Paul, Arch. Ration. Mech. Anal., 223 (2017), pp. 57–94] and [F. Golse and T. Paul, Math. Models Methods Appl. Sci., 32 (2022), pp. 941–963], respectively.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 955-992, February 2024. <br/> Abstract. The Liouville equation with nonconstant magnetic field is obtained as a limit in the Planck constant [math] of the von Neumann equation with the same magnetic field. The convergence is with respect to an appropriate semiclassical pseudodistance, and consequently with respect to the Monge–Kantorovich distance. Uniform estimates both in [math] and [math] are proved for the specific 2D case of a magnetic vector potential of the form [math]. As an application, an observation inequality for the von Neumann equation with a magnetic vector potential is obtained. These results are a magnetic variant of the works [F. Golse and T. Paul, Arch. Ration. Mech. Anal., 223 (2017), pp. 57–94] and [F. Golse and T. Paul, Math. Models Methods Appl. Sci., 32 (2022), pp. 941–963], respectively.
The Magnetic Liouville Equation as a Semiclassical Limit
10.1137/22M1528562
SIAM Journal on Mathematical Analysis
2024-01-19T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Immanuel Ben Porat
The Magnetic Liouville Equation as a Semiclassical Limit
56
1
955
992
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1528562
https://epubs.siam.org/doi/abs/10.1137/22M1528562?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Asymptotic Behavior of Solutions to the Cauchy Problem for 1D [math]-System with Space-Dependent Damping
https://epubs.siam.org/doi/abs/10.1137/23M1588937?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 993-1015, February 2024. <br/> Abstract. We consider the Cauchy problem for a one-dimensional [math]-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangian coordinate. Our concern is an asymptotic behavior of solutions, which is expected to be the diffusion wave based on the Darcy law. In fact, in the constant coefficient case Hsiao and Liu [Comm. Math. Phys., 143 (1992), pp. 599–605] showed the asymptotic behavior under suitable smallness conditions for the first time. After this work, there has been much literature, but there are few works that focus on the space-dependent damping case, as far as we know. In this paper we treat this space-dependent case, as a first step when the coefficient is around some positive constant.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 993-1015, February 2024. <br/> Abstract. We consider the Cauchy problem for a one-dimensional [math]-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangian coordinate. Our concern is an asymptotic behavior of solutions, which is expected to be the diffusion wave based on the Darcy law. In fact, in the constant coefficient case Hsiao and Liu [Comm. Math. Phys., 143 (1992), pp. 599–605] showed the asymptotic behavior under suitable smallness conditions for the first time. After this work, there has been much literature, but there are few works that focus on the space-dependent damping case, as far as we know. In this paper we treat this space-dependent case, as a first step when the coefficient is around some positive constant.
Asymptotic Behavior of Solutions to the Cauchy Problem for 1D [math]-System with Space-Dependent Damping
10.1137/23M1588937
SIAM Journal on Mathematical Analysis
2024-01-25T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Akitaka Matsumura
Kenji Nishihara
Asymptotic Behavior of Solutions to the Cauchy Problem for 1D [math]-System with Space-Dependent Damping
56
1
993
1015
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1588937
https://epubs.siam.org/doi/abs/10.1137/23M1588937?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise
https://epubs.siam.org/doi/abs/10.1137/22M1544440?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1016-1067, February 2024. <br/> Abstract. This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations are established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that [math] is compactly embedded in [math] with [math]. Moreover, the uniqueness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of stochastic PDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee–Infante equations.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1016-1067, February 2024. <br/> Abstract. This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations are established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that [math] is compactly embedded in [math] with [math]. Moreover, the uniqueness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of stochastic PDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee–Infante equations.
Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise
10.1137/22M1544440
SIAM Journal on Mathematical Analysis
2024-01-29T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Jiaohui Xu
Tomás Caraballo
José Valero
Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise
56
1
1016
1067
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1544440
https://epubs.siam.org/doi/abs/10.1137/22M1544440?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects
https://epubs.siam.org/doi/abs/10.1137/22M1495937?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1068-1113, February 2024. <br/> Abstract. In this paper we derive a two dimensional spray model with gyroscopic effects as the mean-field limit of a system modeling the interaction between an incompressible fluid and a finite number of solid particles. This spray model has been studied by Moussa and Sueur in [Asymptot. Anal., 81 (2013), pp. 53–91]; in particular, the mean-field limit was established in the case of [math] interactions. First, we prove the local in time existence and uniqueness of strong solutions of a monokinetic version of the model with a fixed point method. Then we adapt the proof of Duerinckx and Serfaty [Duke Math. J., 169 (2020), pp. 2887–2935] to establish the mean-field limit to the spray model in the monokinetic regime in the case of Coulomb interactions.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1068-1113, February 2024. <br/> Abstract. In this paper we derive a two dimensional spray model with gyroscopic effects as the mean-field limit of a system modeling the interaction between an incompressible fluid and a finite number of solid particles. This spray model has been studied by Moussa and Sueur in [Asymptot. Anal., 81 (2013), pp. 53–91]; in particular, the mean-field limit was established in the case of [math] interactions. First, we prove the local in time existence and uniqueness of strong solutions of a monokinetic version of the model with a fixed point method. Then we adapt the proof of Duerinckx and Serfaty [Duke Math. J., 169 (2020), pp. 2887–2935] to establish the mean-field limit to the spray model in the monokinetic regime in the case of Coulomb interactions.
Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects
10.1137/22M1495937
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Matthieu Ménard
Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects
56
1
1068
1113
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1495937
https://epubs.siam.org/doi/abs/10.1137/22M1495937?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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An Optimal Transport Analogue of the Rudin–Osher–Fatemi Model and Its Corresponding Multiscale Theory
https://epubs.siam.org/doi/abs/10.1137/23M1564109?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1114-1148, February 2024. <br/> Abstract. In the first part of this paper we develop a theory for image restoration with a learned regularizer that is analogous to that of Meyer’s geometric characterization of solutions of the classical variational method of Rudin–Osher–Fatemi (ROF). The learned regularizer we use is a Kantorovich potential for an optimal transport problem of mapping a distribution of noisy images onto clean ones, as first proposed by Lunz, Öktem, and Schönlieb. We show that the effect of their restoration method on the distribution of the images is an explicit Euler discretization of a gradient flow on probability space, while our variational problem, dubbed Wasserstein ROF (WROF), is the corresponding implicit Euler discretization. We obtain our geometric characterization of the solution in this learned regularizer setting by first proving a much more general convex analysis theorem for variational problems having solutions characterized by projections. We then use optimal transport arguments to obtain the corresponding theorem for WROF from this general result, as well as a natural decomposition of a transport map into large scale “features” and small scale “details,” where scale refers to the magnitude of the transport distance. In the second part of the paper we leverage our theory for restoration with learned regularizers to analyze two algorithms which iterate WROF. We refer to these as iterative regularization and multiscale transport. For the former we obtain a proof of convergence to the clean data. For the latter we produce successive approximations to the target distribution that match it up to finer and finer scales. These two algorithms are in complete analogy to well-known effective methods based on ROF for iterative denoising, respectively hierarchical image decomposition. We also obtain an analogue of the Tadmor–Nezzar–Vese energy identity, which decomposes the Wasserstein 2 distance between two measures into a sum of nonnegative terms that correspond to transport costs at different scales.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1114-1148, February 2024. <br/> Abstract. In the first part of this paper we develop a theory for image restoration with a learned regularizer that is analogous to that of Meyer’s geometric characterization of solutions of the classical variational method of Rudin–Osher–Fatemi (ROF). The learned regularizer we use is a Kantorovich potential for an optimal transport problem of mapping a distribution of noisy images onto clean ones, as first proposed by Lunz, Öktem, and Schönlieb. We show that the effect of their restoration method on the distribution of the images is an explicit Euler discretization of a gradient flow on probability space, while our variational problem, dubbed Wasserstein ROF (WROF), is the corresponding implicit Euler discretization. We obtain our geometric characterization of the solution in this learned regularizer setting by first proving a much more general convex analysis theorem for variational problems having solutions characterized by projections. We then use optimal transport arguments to obtain the corresponding theorem for WROF from this general result, as well as a natural decomposition of a transport map into large scale “features” and small scale “details,” where scale refers to the magnitude of the transport distance. In the second part of the paper we leverage our theory for restoration with learned regularizers to analyze two algorithms which iterate WROF. We refer to these as iterative regularization and multiscale transport. For the former we obtain a proof of convergence to the clean data. For the latter we produce successive approximations to the target distribution that match it up to finer and finer scales. These two algorithms are in complete analogy to well-known effective methods based on ROF for iterative denoising, respectively hierarchical image decomposition. We also obtain an analogue of the Tadmor–Nezzar–Vese energy identity, which decomposes the Wasserstein 2 distance between two measures into a sum of nonnegative terms that correspond to transport costs at different scales.
An Optimal Transport Analogue of the Rudin–Osher–Fatemi Model and Its Corresponding Multiscale Theory
10.1137/23M1564109
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Tristan Milne
Adrian Nachman
An Optimal Transport Analogue of the Rudin–Osher–Fatemi Model and Its Corresponding Multiscale Theory
56
1
1114
1148
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1564109
https://epubs.siam.org/doi/abs/10.1137/23M1564109?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Shearing Viscoelasticity in Partially Dissipative Timoshenko–Boltzmann Systems
https://epubs.siam.org/doi/abs/10.1137/23M1568375?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1149-1178, February 2024. <br/> Abstract. We investigate both the mathematical modeling and stability methods for a new integro-differential system referred to as the viscoelastic Timoshenko–Boltzmann model. The modeling is developed for materials with hereditary memory under the creation time scenario whose foundation goes back to Boltzmann’s superposition principle in linear viscoelasticity, complemented by Timoshenko’s assumptions concerning shearing in certain beam vibrations. The mathematical methodology provides a comprehensive characterization of the uniform stability for the partially damped Timoshenko–Boltzmann system through the identity of wave speeds on the structural coefficients and a pointwise dissipative condition on the memory kernel that does not require differential inequalities.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1149-1178, February 2024. <br/> Abstract. We investigate both the mathematical modeling and stability methods for a new integro-differential system referred to as the viscoelastic Timoshenko–Boltzmann model. The modeling is developed for materials with hereditary memory under the creation time scenario whose foundation goes back to Boltzmann’s superposition principle in linear viscoelasticity, complemented by Timoshenko’s assumptions concerning shearing in certain beam vibrations. The mathematical methodology provides a comprehensive characterization of the uniform stability for the partially damped Timoshenko–Boltzmann system through the identity of wave speeds on the structural coefficients and a pointwise dissipative condition on the memory kernel that does not require differential inequalities.
Shearing Viscoelasticity in Partially Dissipative Timoshenko–Boltzmann Systems
10.1137/23M1568375
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Eduardo H. Gomes Tavares
Marcio A. Jorge Silva
To Fu Ma
Higidio P. Oquendo
Shearing Viscoelasticity in Partially Dissipative Timoshenko–Boltzmann Systems
56
1
1149
1178
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1568375
https://epubs.siam.org/doi/abs/10.1137/23M1568375?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Two Slope Functions Minimizing Fractional Seminorms and Applications to Misfit Dislocations
https://epubs.siam.org/doi/abs/10.1137/23M1568910?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1179-1196, February 2024. <br/> Abstract. We consider periodic piecewise affine functions, defined on the real line, with two given slopes, one positive and one negative, and prescribed length scale of the intervals where the slope is negative. We prove that, in such a class, the minimizers of [math]-fractional Gagliardo seminorm densities, with [math], are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals, where the slope is constant, vanishes. Our results, for [math], have relevant applications to the van der Merwe theory of misfit dislocations at semicoherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semicoherent limit as the ratio between the two lattice spacings tends to one.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1179-1196, February 2024. <br/> Abstract. We consider periodic piecewise affine functions, defined on the real line, with two given slopes, one positive and one negative, and prescribed length scale of the intervals where the slope is negative. We prove that, in such a class, the minimizers of [math]-fractional Gagliardo seminorm densities, with [math], are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals, where the slope is constant, vanishes. Our results, for [math], have relevant applications to the van der Merwe theory of misfit dislocations at semicoherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semicoherent limit as the ratio between the two lattice spacings tends to one.
Two Slope Functions Minimizing Fractional Seminorms and Applications to Misfit Dislocations
10.1137/23M1568910
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Lucia De Luca
Marcello Ponsiglione
Emanuele Spadaro
Two Slope Functions Minimizing Fractional Seminorms and Applications to Misfit Dislocations
56
1
1179
1196
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1568910
https://epubs.siam.org/doi/abs/10.1137/23M1568910?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction
https://epubs.siam.org/doi/abs/10.1137/23M1569885?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1197-1222, February 2024. <br/> Abstract. We study the Murray adaptation of the Noyes–Field five-step model of the Belousov–Zhabotinsky (BZ) reaction in the case when a tuning parameter [math], which determines the level of the bromide ion far ahead of the propagating wave, is bigger than 1 and when the delay in generation of the bromous acid is taken into account. The existence of wavefronts in the delayed BZ system was previously established only in the monostable situation with [math], the physically relevant bistable situation where [math] (in real experiments, [math] varies between 5 and 50) was left open. We complete the study by showing that the BZ system with [math] admits monotone traveling fronts. Note that one of the stable equilibria of the BZ model is not isolated. This circumstance does not allow the direct application of the topological or analytical methods previously elaborated for the analysis of the existence of bistable waves.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1197-1222, February 2024. <br/> Abstract. We study the Murray adaptation of the Noyes–Field five-step model of the Belousov–Zhabotinsky (BZ) reaction in the case when a tuning parameter [math], which determines the level of the bromide ion far ahead of the propagating wave, is bigger than 1 and when the delay in generation of the bromous acid is taken into account. The existence of wavefronts in the delayed BZ system was previously established only in the monostable situation with [math], the physically relevant bistable situation where [math] (in real experiments, [math] varies between 5 and 50) was left open. We complete the study by showing that the BZ system with [math] admits monotone traveling fronts. Note that one of the stable equilibria of the BZ model is not isolated. This circumstance does not allow the direct application of the topological or analytical methods previously elaborated for the analysis of the existence of bistable waves.
Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction
10.1137/23M1569885
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Karel Hasík
Jana Kopfová
Petra Nábělková
Sergei Trofimchuk
Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction
56
1
1197
1222
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1569885
https://epubs.siam.org/doi/abs/10.1137/23M1569885?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Global [math] Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Divergence Form
https://epubs.siam.org/doi/abs/10.1137/22M1512120?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1223-1263, February 2024. <br/> Abstract. We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for the kinetic Kolmogorov–Fokker–Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable [math] and satisfy a vanishing mean oscillation (VMO) type condition. We consider the [math] case separately and treat more general equations, which include the relativistic KFP equation. This paper is a continuation of our previous work on [math] estimates for KFP equations in nondivergence form.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1223-1263, February 2024. <br/> Abstract. We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for the kinetic Kolmogorov–Fokker–Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable [math] and satisfy a vanishing mean oscillation (VMO) type condition. We consider the [math] case separately and treat more general equations, which include the relativistic KFP equation. This paper is a continuation of our previous work on [math] estimates for KFP equations in nondivergence form.
Global [math] Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Divergence Form
10.1137/22M1512120
SIAM Journal on Mathematical Analysis
2024-01-31T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Hongjie Dong
Timur Yastrzhembskiy
Global [math] Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Divergence Form
56
1
1223
1263
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1512120
https://epubs.siam.org/doi/abs/10.1137/22M1512120?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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A Regularity Theory for Parabolic Equations with Anisotropic Nonlocal Operators in [math] Spaces
https://epubs.siam.org/doi/abs/10.1137/23M1574944?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1264-1299, February 2024. <br/> Abstract. In this paper, we present an [math]-regularity theory for parabolic equations of the form [math] Here, [math] represents anisotropic nonlocal operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: [math] To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón–Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic nonlocal operators and parabolic equations with isotropic nonlocal operators.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1264-1299, February 2024. <br/> Abstract. In this paper, we present an [math]-regularity theory for parabolic equations of the form [math] Here, [math] represents anisotropic nonlocal operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: [math] To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón–Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic nonlocal operators and parabolic equations with isotropic nonlocal operators.
A Regularity Theory for Parabolic Equations with Anisotropic Nonlocal Operators in [math] Spaces
10.1137/23M1574944
SIAM Journal on Mathematical Analysis
2024-02-06T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Jae-Hwan Choi
Jaehoon Kang
Daehan Park
A Regularity Theory for Parabolic Equations with Anisotropic Nonlocal Operators in [math] Spaces
56
1
1264
1299
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1574944
https://epubs.siam.org/doi/abs/10.1137/23M1574944?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics
https://epubs.siam.org/doi/abs/10.1137/23M1549948?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1300-1320, February 2024. <br/> Abstract. We are concerned with the isothermal limit of entropy solutions in [math], containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in [math] of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent [math]. This is achieved by combining careful entropy analysis and refined kinetic formulation with a compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent [math]. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to [math]. The explicit asymptotic analysis of the Riemann solutions containing the vacuum states is also presented.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1300-1320, February 2024. <br/> Abstract. We are concerned with the isothermal limit of entropy solutions in [math], containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in [math] of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent [math]. This is achieved by combining careful entropy analysis and refined kinetic formulation with a compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent [math]. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to [math]. The explicit asymptotic analysis of the Riemann solutions containing the vacuum states is also presented.
Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics
10.1137/23M1549948
SIAM Journal on Mathematical Analysis
2024-02-08T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Gui-Qiang G. Chen
Fei-Min Huang
Tian-Yi Wang
Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics
56
1
1300
1320
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1549948
https://epubs.siam.org/doi/abs/10.1137/23M1549948?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials
https://epubs.siam.org/doi/abs/10.1137/23M1560148?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1321-1426, February 2024. <br/> Abstract. We prove the well-posedness for the noncutoff Boltzmann equation with soft potentials when the initial datum is close to the global Maxwellian and has only polynomial decay at the large velocities in [math] space. As a result, we get the propagation of the exponential moments and the sharp rates of the convergence to the global Maxwellian which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in [math] space.
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1321-1426, February 2024. <br/> Abstract. We prove the well-posedness for the noncutoff Boltzmann equation with soft potentials when the initial datum is close to the global Maxwellian and has only polynomial decay at the large velocities in [math] space. As a result, we get the propagation of the exponential moments and the sharp rates of the convergence to the global Maxwellian which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in [math] space.
Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials
10.1137/23M1560148
SIAM Journal on Mathematical Analysis
2024-02-09T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Chuqi Cao
Ling-Bing He
Jie Ji
Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials
56
1
1321
1426
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/23M1560148
https://epubs.siam.org/doi/abs/10.1137/23M1560148?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds
https://epubs.siam.org/doi/abs/10.1137/22M1529312?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1427-1458, February 2024. <br/> Abstract. In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) [math], [math], [math]. The torus component [math] can be generalized to [math]-dimensional compact manifolds [math]. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher-dimensional analogue and the focusing scenario, and give some further remarks on this research line. This result can be regarded as the waveguide analogue of [C. Miao, G. Xu, and L. Zhao, J. Differential Equations, 251 (2011), pp. 3381–3402], [B. Pausader, Dyn. Partial Differ. Equ., 4 (2007), pp. 197–225], [B. Pausader, J. Funct. Anal., 256 (2009), pp. 2473–2517], [B. Pausader, Discrete Contin. Dyn. Syst., 24 (2009), pp. 1275–1292] and the 4NLS analogue of Tzvetkov and Visciglia [N. Tzvetkov and N. Visciglia, Rev. Mat. Iberoam., 32 (2016), pp. 1163–1188].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1427-1458, February 2024. <br/> Abstract. In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) [math], [math], [math]. The torus component [math] can be generalized to [math]-dimensional compact manifolds [math]. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher-dimensional analogue and the focusing scenario, and give some further remarks on this research line. This result can be regarded as the waveguide analogue of [C. Miao, G. Xu, and L. Zhao, J. Differential Equations, 251 (2011), pp. 3381–3402], [B. Pausader, Dyn. Partial Differ. Equ., 4 (2007), pp. 197–225], [B. Pausader, J. Funct. Anal., 256 (2009), pp. 2473–2517], [B. Pausader, Discrete Contin. Dyn. Syst., 24 (2009), pp. 1275–1292] and the 4NLS analogue of Tzvetkov and Visciglia [N. Tzvetkov and N. Visciglia, Rev. Mat. Iberoam., 32 (2016), pp. 1163–1188].
Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds
10.1137/22M1529312
SIAM Journal on Mathematical Analysis
2024-02-12T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Xueying Yu
Haitian Yue
Zehua Zhao
Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds
56
1
1427
1458
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1529312
https://epubs.siam.org/doi/abs/10.1137/22M1529312?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics
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Integrability Propagation for a Boltzmann System Describing Polyatomic Gas Mixtures
https://epubs.siam.org/doi/abs/10.1137/22M1539897?ai=s2&mi=3bfys9&af=R
SIAM Journal on Mathematical Analysis, <a href="https://epubs.siam.org/toc/sjmaah/56/1">Volume 56, Issue 1</a>, Page 1459-1494, February 2024. <br/> Abstract. This paper explores the [math] Lebesgue’s integrability propagation, [math], of a system of space homogeneous Boltzmann equations modelling a multicomponent mixture of polyatomic gases based on the continuous internal energy. For typical collision kernels proposed in the literature, [math] moment-entropy-based estimates for the collision operator gain part and a lower bound for the loss part are performed, leading to a vector valued inequality for the collision operator and, consequently, to a differential inequality for the vector valued solutions of the system. This allows one to prove the propagation property of the polynomially weighted [math] norms associated to the vector valued solution of the system of Boltzmann equations. The case [math] is found as a limit of the case [math].
SIAM Journal on Mathematical Analysis, Volume 56, Issue 1, Page 1459-1494, February 2024. <br/> Abstract. This paper explores the [math] Lebesgue’s integrability propagation, [math], of a system of space homogeneous Boltzmann equations modelling a multicomponent mixture of polyatomic gases based on the continuous internal energy. For typical collision kernels proposed in the literature, [math] moment-entropy-based estimates for the collision operator gain part and a lower bound for the loss part are performed, leading to a vector valued inequality for the collision operator and, consequently, to a differential inequality for the vector valued solutions of the system. This allows one to prove the propagation property of the polynomially weighted [math] norms associated to the vector valued solution of the system of Boltzmann equations. The case [math] is found as a limit of the case [math].
Integrability Propagation for a Boltzmann System Describing Polyatomic Gas Mixtures
10.1137/22M1539897
SIAM Journal on Mathematical Analysis
2024-02-15T08:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ricardo Alonso
Milana Čolić
Integrability Propagation for a Boltzmann System Describing Polyatomic Gas Mixtures
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1
1459
1494
2024-02-29T08:00:00Z
2024-02-29T08:00:00Z
10.1137/22M1539897
https://epubs.siam.org/doi/abs/10.1137/22M1539897?ai=s2&mi=3bfys9&af=R
© 2024 Society for Industrial and Applied Mathematics