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2012

Volume 44, Issue 1, pp. 1-540


Dynamics of Soliton-like Solutions for Slowly Varying, Generalized KdV Equations: Refraction versus Reflection

Claudio Muñoz

SIAM J. Math. Anal. 44, pp. 1-60 (60 pages)

Online Publication Date: January 05, 2012

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In this work we continue our study of the description of the soliton-like solutions of the variable coefficients, subcritical gKdV equation $u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0$, in $\mathbb{R}_t\times \mathbb{R}_x$, $m=2,3$, and 4, with $0\leq \lambda<1$, $1<a(\cdot )<2$ a strictly increasing, positive, and asymptotically flat potential, and $\varepsilon$ small enough. In [C. Muñoz, “On the soliton dynamics under a slowly varying medium for generalized KdV equations,” Anal. PDE, to appear] we proved the existence (and uniqueness in most cases) of a pure soliton-like solution $u(t)$ satisfying $\lim_{t\to -\infty}\|u(t) - Q(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0$, $0\leq \lambda<1$, provided $\varepsilon$ is small enough. Here $R(t,x) := Q_c(x-(c-\lambda)t)$ is the standard $H^1$-soliton solution of $R_t + (R_{xx} -\lambda R + R^m)_x =0$. In addition, this solution is global in time and satisfies (i) for all $0<\lambda\leq\frac{5-m}{m+3}$, $\sup_{t\gg \frac 1\varepsilon }\|u(t) - 2^{-1/(m-1)}Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})} \leq K\varepsilon^{1/2}$, for suitable scaling and translation parameters $c_\infty(\lambda)\geq 1$ and $\rho'(t) \sim (c_\infty-\lambda)$, and for $K>0$. In the cubic case, $m=3$, this result also holds for $\lambda=0$. The purpose of this paper is the following: We give an almost complete description of the remaining case $\frac{5-m}{m+3}<\lambda<1$. Surprisingly, there exists a fixed, positive number $\tilde \lambda \in (\frac{5-m}{m+3} ,1)$, independent of $\varepsilon$, such that the following alternative holds: (1) Refraction. For all $\frac{5-m}{m+3}<\lambda<\tilde \lambda$, the soliton solution behaves as in [C. Muñoz, “On the soliton dynamics under a slowly varying medium for generalized KdV equations,” Anal. PDE, to appear] and satisfies (i) above, but now $\lambda <c_\infty<1$ and $\rho'(t) \sim c_\infty -\lambda >0$. (2) Reflection. If $\tilde \lambda <\lambda<1$, then the soliton-like solution is reflected by the potential and satisfies $\sup_{t\gg \frac 1\varepsilon }\|u(t) - Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})} \leq K\varepsilon^{1/2},$ with $0<c_\infty <\lambda$ and $\rho'(t) \sim c_\infty-\lambda <0$. This last is a completely new type of soliton-like solution for gKdV equations, also present in the nonlinear Schrödinger case [C. Muñoz, “On the soliton dynamics under slowly varying medium for generalized nonlinear Schrödinger equations,” Math. Ann., to appear]. Moreover, for any $0<\lambda<1$, with $\tilde\lambda\neq \lambda$, the solution is not pure as $t\to +\infty$, in the sense that $\limsup_{t\to +\infty}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot -\rho(t)) \|_{H^1(\mathbb{R})}>0,$ with $\kappa(\lambda) $ depending on $\lambda$.

New Examples Satisfying Ma–Trudinger–Wang Conditions

Paul W. Y. Lee and Jiayong Li

SIAM J. Math. Anal. 44, pp. 61-73 (13 pages)

Online Publication Date: January 05, 2012

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In this paper, we study the Ma–Trudinger–Wang (MTW) conditions for cost functions $c$ which are of the form $c=l\circ d$, where $d$ is a Riemannian distance function with constant sectional curvature. In this case, the MTW conditions are equivalent to some computable conditions on the function $l$. As a corollary, we give some new costs on Riemannian manifolds of constant negative curvature for which the MTW conditions are satisfied.

Dynamics of Compressible Non-isothermal Fluids of Non-Newtonian Korteweg Type

Matthias Kotschote

SIAM J. Math. Anal. 44, pp. 74-101 (28 pages)

Online Publication Date: January 13, 2012

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The equations of motion for compressible fluids of Korteweg type as derived by Dunn and Serrin in 1985 are studied in their full generality: the Korteweg tensor is assumed to be an arbitrary function of the form $\mathcal{K} := \left( - \rho^2 \partial_{\rho} \psi + \rho \nabla \cdot ( \kappa \nabla \rho) \right) \mathcal{I} - \kappa \nabla \rho \otimes \nabla \rho, \quad \kappa := 2 \rho \partial_{\phi} \psi(\rho,\theta,\phi), \quad \phi:=|\nabla \rho|^2,$ where $\psi$ denotes Helmholtz free energy density and the capillarity $\kappa$ is subject only to the natural positivity conditions $\kappa(\rho,\theta,\phi) >0, \quad \kappa(\rho,\theta,\phi) + 2 \phi \partial_{\phi} \kappa(\rho,\theta,\phi) > 0, \quad \rho, \theta, \phi \ge 0.$ The viscous stress is supposed to be of generalized Newtonian type. The main result of the paper establishes well-posedness on domains with compact boundaries; the proof is based on refined methods of maximal regularity.

The Cauchy Problem on the Compressible Two-fluids Euler–Maxwell Equations

Renjun Duan, Qingqing Liu, and Changjiang Zhu

SIAM J. Math. Anal. 44, pp. 102-133 (32 pages)

Online Publication Date: January 13, 2012

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In this paper, we are concerned with the Cauchy problem on the compressible isentropic two-fluids Euler–Maxwell equations in three dimensions. The global existence of solutions near constant steady states with the vanishing electromagnetic field is established, and the time-decay rates of perturbed solutions in $L^q$ space for $2\leq q\leq \infty$ are obtained. The proof for existence is due to the classical energy method, and the investigation of large-time behavior is based on linearized analysis of one-fluid Euler–Maxwell equations and damped Euler equations. As a byproduct of our approach, some time-decay rates obtained by Sideris, Thomases, and Wang [Comm. Partial Differential Equations, 28 (2003), pp. 795–816] for the nonlinear damped Euler system are improved.

Inverse Transport with Isotropic Time-harmonic Sources

Guillaume Bal and François Monard

SIAM J. Math. Anal. 44, pp. 134-161 (28 pages)

Online Publication Date: January 13, 2012

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This paper concerns the reconstruction of the scattering coefficient in a two-dimensional transport equation from angularly averaged measurements when the probing source is isotropic and time-harmonic. This is a practical setting in the medical imaging modality called optical tomography. As the modulation frequency of the source increases, we show that the reconstruction of the scattering coefficient improves. More precisely, as the frequency $\omega$ increases, we show that all frequencies of the scattering coefficient lower than $b$ are reconstructed stably with an accuracy that improves as $\omega$ increases and $b$ decreases. The proofs are based on an analysis of the single scattering singularities of the transport equation and on careful analyses of oscillatory integrals by stationary phase arguments.

Interpolatory Multiscale Representation for Functions between Manifolds

Andreas Weinmann

SIAM J. Math. Anal. 44, pp. 162-191 (30 pages)

Online Publication Date: January 13, 2012

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We investigate interpolatory multiscale transformations for functions between manifolds which are based on interpolatory subdivision rules. We characterize the Hölder–Zygmund smoothness of a function between manifolds in terms of the coefficient decay w.r.t. this multiscale transform.

A Singular Limit for Compressible Rotating Fluids

Eduard Feireisl, Isabelle Gallagher, and Antonín Novotný

SIAM J. Math. Anal. 44, pp. 192-205 (14 pages)

Online Publication Date: January 13, 2012

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We consider a singular limit problem for the Navier–Stokes system of a rotating compressible fluid, where the Rossby and Mach numbers tend simultaneously to zero. The limit problem is identified as the two-dimensional (2D) Navier–Stokes system in the “horizontal” variables containing an extra term that accounts for compressibility in the original system.

Bifurcation from Semitrivial Standing Waves and Ground States for a System of Nonlinear Schrödinger Equations

Mathieu Colin and Masahito Ohta

SIAM J. Math. Anal. 44, pp. 206-223 (18 pages)

Online Publication Date: January 13, 2012

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We consider a system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability and instability of standing waves bifurcating from the semitrivial standing wave of the system. The stability and instability of the semitrivial standing wave at the bifurcation point are also studied. Moreover, we determine the set of the ground states completely.

Ergodicity of Stochastic Curve Shortening Flow in the Plane

Abdelhadi Es-Sarhir and Max-K. von Renesse

SIAM J. Math. Anal. 44, pp. 224-244 (21 pages)

Online Publication Date: January 19, 2012

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We study models of the motion by mean curvature of an $(1+1)$-dimensional interface with random forcing. For the well-posedness we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov–Rozovskiĭ, replacing the standard coercivity assumption by a Lyapunov-type condition. We also study the long-term behavior, showing that the homogeneous normal noise model [N. Dirr, S. Luckhaus, and M. Novaga, Calc. Var. Partial Differential Equations, 13 (2001), pp. 405–425], [P. E. Souganidis and N. K. Yip, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 1–23] with periodic boundary conditions converges to a spatially constant profile whose height behaves like a Brownian motion. For the additive vertical noise model with Dirichlet boundary conditions we show ergodicity, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek [Ann. Probab., 38 (2010), pp. 1401–1443].

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

J.-F. Babadjian, G. A. Francfort, and M. G. Mora

SIAM J. Math. Anal. 44, pp. 245-292 (48 pages)

Online Publication Date: January 19, 2012

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Nonassociative elasto-plasticity is the working model of plasticity for soil and rock mechanics. Yet, it is usually viewed as nonvariational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.

Global Exponential Convergence to Variational Traveling Waves in Cylinders

C. B. Muratov and M. Novaga

SIAM J. Math. Anal. 44, pp. 293-315 (23 pages)

Online Publication Date: January 19, 2012

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We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg–Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for “pushed” fronts.

Strong Well-posedness of a Diffuse Interface Model for a Viscous, Quasi-incompressible Two-phase Flow

Helmut Abels

SIAM J. Math. Anal. 44, pp. 316-340 (25 pages)

Online Publication Date: January 26, 2012

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In this article we discuss the existence of strong solutions locally in time for a model of a binary mixture of viscous incompressible fluids in a bounded domain. The model was derived by Lowengrub and Truskinovski. It is used to describe a diffuse interface model for a two-phase flow of two viscous incompressible Newtonian fluids with different densities. The fluids are macroscopically immiscible but partially mix in a small interfacial region. The model leads to a system of Navier–Stokes/Cahn–Hilliard type. Using a suitable result on maximal $L^2$-regularity for the linearized system, the existence of strong solutions is shown with the aid of the contraction mapping principle. The analysis shows that in the case of different densities the system is coupled in highest order and the principal part of the linearized system is of very different structure compared to the case of same densities. The linear system is solved with the aid of a general result on an abstract damped wave equation by Chen and Triggiani.

Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators

John Sylvester

SIAM J. Math. Anal. 44, pp. 341-354 (14 pages)

Online Publication Date: February 02, 2012

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Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled $2 \times 2$ system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite-dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.

Limiting Absorption Principle and Perfectly Matched Layer Method for Dirichlet Laplacians in Quasi-cylindrical Domains

Victor Kalvin

SIAM J. Math. Anal. 44, pp. 355-382 (28 pages)

Online Publication Date: February 09, 2012

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We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.

Stability of the Scattering from a Large Electromagnetic Cavity in Two Dimensions

Gang Bao, KiHyun Yun, and Zhengfang Zhou

SIAM J. Math. Anal. 44, pp. 383-404 (22 pages)

Online Publication Date: February 09, 2012

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This work is concerned with a time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity embedded in the infinite ground plane. Because of the highly oscillatory nature of the solution for large or deep cavity, the model scattering problem is challenging both mathematically and computationally. A variational formulation reduces the scattering problem into a bounded domain (the cavity) problem. The stability of the solution is established for the bounded domain problem in the energy space. Moreover, our stability estimates provide the explicit dependence on the high wave number and the depth of the cavity.

Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

Haitao Fan and Xiao-Biao Lin

SIAM J. Math. Anal. 44, pp. 405-436 (32 pages)

Online Publication Date: February 14, 2012

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We study the existence of standing waves for liquid/vapor phase transition in a spherically symmetric nozzle. The system is singularly perturbed and the solution consists of an internal layer where the liquid quickly becomes vapor. Using methods from dynamical systems theory, we prove the existence of the internal layer as a heteroclinic orbit connecting the liquid state to the vapor state. The heteroclinic orbit is reproduced numerically and is also shown numerically to be a transversal heteroclinic orbit. The proof of the existence of an exact standing wave solution near the singular limit is based on the geometric singular perturbation theory and is outlined in the paper.

A Centennial of the Zaremba–Hopf–Oleinik Lemma

A. I. Nazarov

SIAM J. Math. Anal. 44, pp. 437-453 (17 pages)

Online Publication Date: February 16, 2012

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We consider the Hopf–Oleinik normal derivative lemma for elliptic and parabolic equations under minimal restrictions on lower-order coefficients. Boundary gradient estimates for solutions are also established.

Optimal Transport with Branching Distance Costs and the Obstacle Problem

Fabio Cavalletti

SIAM J. Math. Anal. 44, pp. 454-482 (29 pages)

Online Publication Date: February 23, 2012

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We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along a family of geodesics. We introduce three assumptions on a given $d_{N}$-monotone transference plan $\pi$ which imply, respectively, strong consistency of disintegration, continuity of the conditional probabilities of the first marginal, and a regularity property for the geometry of chain of transport rays. We show that this regularity is sufficient for the construction of a transport map with the same transport cost of $\pi$. We apply these results to the Monge problem in $\mathbb{R}^{d}$ with smooth, convex, and compact obstacle obtaining the existence of an optimal map, provided the first marginal is absolutely continuous with respect to the $d$-dimensional Lebesgue measure.

Optimal Thin Torsion Rods and Cheeger Sets

Guy Bouchitté, Ilaria Fragalà, Ilaria Lucardesi, and Pierre Seppecher

SIAM J. Math. Anal. 44, pp. 483-512 (30 pages)

Online Publication Date: February 28, 2012

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We carry out an asymptotic analysis of the following shape optimization problem: a given volume fraction of elastic material must be distributed in a cylindrical design region of infinitesimal cross section in order to maximize resistance to a twisting load. We derive a limit rod model written in different equivalent formulations and for which we are able to give necessary and sufficient conditions characterizing optimal configurations. Eventually we show that for a convex design region and for very small volume fractions, the optimal shape tends to concentrate section by section near the boundary of the Cheeger set of the design. These results were announced in [G. Bouchitté, I. Fragalà, and P. Seppecher, C. R. Math., 348 (2010), pp. 467–471].

On Weak Completeness of the Set of Entropy Solutions to a Degenerate Nonlinear Parabolic Equation

Evgeniy Panov

SIAM J. Math. Anal. 44, pp. 513-535 (23 pages)

Online Publication Date: February 28, 2012

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We prove that weak limits of approximate entropy solutions to a one-dimensional degenerate parabolic equation are entropy solutions as well.

Addendum to “Convergence in Strongly Monotone Systems with an Increasing First Integral”

Murad Banaji and David Angeli

SIAM J. Math. Anal. 44, pp. 536-537 (2 pages)

Online Publication Date: February 28, 2012

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We correct an error in the statement of Theorem 2.4 in “Convergence in strongly monotone systems with an increasing first integral” (M. Banaji and D. Angeli, SIAM J. Math. Anal., 42 (2010), pp. 334–353) on global convergence in strongly monotone dynamical systems with an increasing first integral.

Erratum: Global Stability of Monostable Traveling Waves For Nonlocal Time-delayed Reaction-diffusion Equations

Ming Mei, Chunhua Ou, and Xiao-Qiang Zhao

SIAM J. Math. Anal. 44, pp. 538-540 (3 pages)

Online Publication Date: February 28, 2012

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This short note is to fix a gap in the proof of Lemma 3.8 in our paper [M. Mei, C. Ou, and X.-Q. Zhao, SIAM J. Math. Anal., 42 (2010), pp. 2762–2790].
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