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2012

Volume 44, Issue 3 (partial)


Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems

János Karátson

SIAM J. Math. Anal. 44, pp. 1279-1303 (25 pages)

Online Publication Date: May 01, 2012

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The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.

Well-Posedness for a Multidimensional Viscous Liquid-Gas Two-Phase Flow Model

Chengchun Hao and Hai-Liang Li

SIAM J. Math. Anal. 44, pp. 1304-1332 (29 pages)

Online Publication Date: May 01, 2012

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The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.

Inverse Boundary Value Problem for Schrödinger Equation in Two Dimensions

O. Yu. Imanuvilov and M. Yamamoto

SIAM J. Math. Anal. 44, pp. 1333-1339 (7 pages)

Online Publication Date: May 03, 2012

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We relax the regularity condition on potentials of the Schrödinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [O. Imanuvilov, G. Uhlmann, and M. Yamamoto, J. Amer. Math. Soc., 23 (2010), pp. 655–691] and [A. Bukhgeim, J. Inverse Ill-Posed Probl., 16 (2008), pp. 19–34].

Global Bifurcation of Solutions for Crime Modeling Equations

Robert Stephen Cantrell, Chris Cosner, and Raúl Manásevich

SIAM J. Math. Anal. 44, pp. 1340-1358 (19 pages)

Online Publication Date: May 03, 2012

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We study pattern formation in a quasi-linear system of two elliptic equations that was developed by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249–1267] as a model for residential burglary. That model is based on the observation that the rate of burglaries of houses that have been burglarized recently and their close neighbors is typically higher than the average rate in the larger community, which creates hotspots for burglary. The patterns generated by the model describe the location of those hotspots. We prove that the system supports global bifurcation of spatially varying solutions from the spatially constant equilibrium, leading to the formation of spatial patterns. The analysis is based on recent results on global bifurcation in quasi-linear elliptic systems derived by Shi and Wang [J. Differential Equations, 7 (2009), pp. 2788–2812]. We show in some cases that near the bifurcation point the bifurcating spatial patterns are stable.

A Model for the Behavior of Fluid Droplets Based on Mean Curvature Flow

Sebastian Helmensdorfer

SIAM J. Math. Anal. 44, pp. 1359-1371 (13 pages)

Online Publication Date: May 08, 2012

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The authors of [W. D. Ristenpart et al., Nature, 461 (2009), pp. 377–380] have observed the following remarkable phenomenon during their experiments. If two oppositely charged droplets of fluid are close enough, at first they attract each other and eventually touch. Surprisingly after that the droplets are repelled from each other, if the initial strength of the charges is high enough. Otherwise they coalesce and form a big drop, as one might expect. We present a theoretical model for these observations using mean curvature flow. The local asymptotic shape of the touching fluid droplets is that of a double cone, where the angle corresponds to the strength of the initial charges. Our model yields a critical angle for the behavior of the touching droplets, and numerical estimates of this angle agree with the experiments. This shows, contrary to general belief (see [W. D. Ristenpart et al., Nature, 461 (2009), pp. 377–380] and [W. D. Ristenpart et al., Phys. Rev. Lett., 103 (2009), 164502]), that decreasing surface energy can explain the phenomenon. To determine the critical angle within our model, we construct appropriate barriers for the mean curvature flow. In [Comm. Partial Differential Equations, 20 (1995), pp. 1937–1958] Angenent, Chopp, and Ilmanen manage to show the existence of one-sheeted and two-sheeted self-expanding solutions with a sufficiently steep double cone as an initial condition. Furthermore they provide arguments for nonuniqueness even among the one-sheeted solutions. We present a proof for this, yielding a slightly stronger result. Using the one-sheeted self-expanders as barriers, we can determine the critical angle for our model.

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Jan van Neerven, Mark Veraar, and Lutz Weis

SIAM J. Math. Anal. 44, pp. 1372-1414 (43 pages)

Online Publication Date: May 08, 2012

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We prove maximal $L^p$-regularity for the stochastic evolution equation $dU(t) + A U(t)\, dt = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), t\in [0,T]$, $U(0) = u_0$, under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12\pi$ on a space $L^q(\mathcal{O},\mu)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space $\mathsf{D}_A(1-\tfrac1p,p)$ we prove existence of a unique strong solution with trajectories in $ L^p(0,T;\mathsf{D}(A))\cap C([0,T];\mathsf{D}_A(1-\tfrac1p,p)),$ provided the nonlinearities $F:[0,T]\times \mathsf{D}(A)\to L^q(\mathcal{O},\mu)$ and $B:[0,T]\times \mathsf{D}(A) \to \gamma(H,\mathsf{D}(A^{\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier–Stokes equation on a smooth bounded domain $\mathcal{O}\subseteq \mathbb{R}^d$ with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\mathcal{O}))^d$ is shown.

Transport of Charged Particles Under Fast Oscillating Magnetic Fields

Mihai Bostan

SIAM J. Math. Anal. 44, pp. 1415-1447 (33 pages)

Online Publication Date: May 08, 2012

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The energy production through thermonuclear fusion requires the confinement of the plasma within a bounded domain. In most cases, such configurations are obtained by using strong magnetic fields. Several models exist for describing the evolution of a strongly magnetized plasma, e.g., guiding-center approximation and the finite Larmor radius regime. The topic of this paper concerns a different approach leading to plasma confinement. More precisely, we are interested in mathematical models with fast oscillating magnetic fields. We provide rigorous derivations for this kind of model and analyze their properties.

Global Weak Solution for a Heat and Sweat Transport System in Three-Dimensional Fibrous Porous Media with Condensation/Evaporation and Absorption

Buyang Li and Weiwei Sun

SIAM J. Math. Anal. 44, pp. 1448-1473 (26 pages)

Online Publication Date: May 08, 2012

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This paper is concerned with heat and moisture transport in three-dimensional porous textile media with complex phase change, condensation/evaporation, and absorption. The physical process is described as multiphase and multicomponent (air, vapor, water, and heat) flow governed by a system of nonlinear, degenerate, and strongly coupled parabolic equations. In this paper, we prove global existence of a positive weak solution under the physical assumption that the initial and background temperatures are not higher than about $1000$ K.

Stability of Traveling Front Solutions with Algebraic Spatial Decay for Some Autocatalytic Chemical Reaction Systems

Yi Li and Yaping Wu

SIAM J. Math. Anal. 44, pp. 1474-1521 (48 pages)

Online Publication Date: May 08, 2012

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This paper is concerned with the linear and nonlinear asymptotic stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, especially for the typical autocatalytic system with reaction rate $u^qv^p$ when $p>1$ and $q\ge 1$. First, for the autocatalytic systems with equal diffusion rates and with more general initial values, the wave fronts with noncritical speeds are proved to be nonlinearly asymptotically stable in some special polynomially weighted spaces for the case where $p>1$ and $q=1$, and the waves fronts with noncritical speeds are proved to be linearly asymptotically stable for the case where $p>1$ and $q>1$. Second, by applying special transformations and appropriate matrix perturbation theories, we establish some abstract results on the existence and analyticity of the Evans function for the more general linear ODE systems with slow algebraic decaying coefficients. Third, by detailed spectral estimates and by applying our abstract results on the Evans function to the autocatalytic systems for the case where $p>1$ and $q\ge 1$ and when the two diffusion rates are close, we prove that the wave fronts with noncritical speeds are linearly exponentially stable in some exponentially weighted spaces. Finally, for the autocatalytic systems with $p>1$ and $q=1$ it can be shown that if the initial perturbation of the wave in $C_{\rm unif}(R)$ space is small in both the unweighted norm and the exponentially weighted norm, then the perturbation stays small in the unweighted norm and decays exponentially in the exponentially weighted norm; further, we can prove that if the initial perturbation is, in addition, small in the $L^1$ norm, then the perturbation also stays small in the $L_1(R)$ norm and decays algebraically in the $C_{\rm unif}(R)$ norm.

Smoothing Property of the Gain Term of the Boltzmann Collision Operator

Jin-Cheng Jiang

SIAM J. Math. Anal. 44, pp. 1522-1543 (22 pages)

Online Publication Date: May 10, 2012

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In this paper, we prove estimates of the smoothing property of the gain part of the Boltzmann collision operator for the full kernel of the hard sphere model in ${\mathbb R}^3$ which are an improvement over estimates by Lions [J. Math. Kyoto Univ., 34 (1994), pp. 391–427, pp. 429–461] and Wennberg [Comm. Partial Differential Equations, 19 (1994), pp. 2057–2074]. In earlier works for different collision models, this type of smoothing estimate has always come with the compactness assumption [P.-L. Lions, J. Math. Kyoto Univ., 34 (1994), pp. 391–427, 429–461] for both variables of the collision kernel. Now we show that the decaying condition in the angular component which is caused by non-head-on collisions for the hard sphere model can replace the compactness assumption. Namely, no artificial truncation of the angular component at the boundary points is needed to prove the regularizing effect. Also the compactness assumption in relative velocity can be relaxed. For large relative velocity the smoothing estimate can hold in proper weighted function space. For small relative velocity, we can only prove a weaker smoothing estimate.

Blow Up Analysis for Anomalous Granular Gases

Thomas Rey

SIAM J. Math. Anal. 44, pp. 1544-1561 (18 pages)

Online Publication Date: May 10, 2012

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In this article we investigate the long-time behavior of the solutions to the energy-dependent, spatially homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description that dissipates energy during collisions. We assume that the gas is “anomalous," in the sense that energy dissipation increases when temperature decreases. This allows the gas to cool down in finite time. We study existence and uniqueness of blow up profiles for this model, together with the trend toward equilibrium and the associated cooling law, generalizing the classical Haff's law for granular gases. To this end, we investigate the asymptotic behavior of the inelastic Boltzmann equation with and without drift term by introducing new strongly “nonlinear” self-similar variables.

Asymptotic Behavior of a Class of Nonlinear Stochastic Heat Equations with Memory Effects

Stefano Bonaccorsi, Giuseppe Da Prato, and Luciano Tubaro

SIAM J. Math. Anal. 44, pp. 1562-1587 (26 pages)

Online Publication Date: May 10, 2012

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In this paper we investigate a class of semilinear stochastic Volterra equations which arise in the theory of heat conduction with memory effects, with dissipative nonlinearities and an additive stochastic term which models a rapidly varying external heat source. We first prove that the problem has a unique solution for all times; further, we analyze the asymptotic behavior of the solution and we prove the existence of an ergodic invariant measure.

Analysis of Oscillations in a Drainage Equation

C. M. Cuesta and J. J. L. Velázquez

SIAM J. Math. Anal. 44, pp. 1588-1616 (29 pages)

Online Publication Date: May 10, 2012

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We analyze rigorously the drainage equation $(\frac{d^3\Phi}{d\tau^3} +1)\Phi^3 = 1$. It is known that all solutions that do not satisfy $\Phi\to 1$ as $\tau \to \infty$ are oscillatory. They oscillate with increasing amplitude as $\tau \to \infty$. We state and make rigorous the precise asymptotic description of these oscillations.

Asymptotic Analysis of Nonsymmetric Linear Operators via $\Gamma$-Convergence

Nadia Ansini and Caterina Ida Zeppieri

SIAM J. Math. Anal. 44, pp. 1617-1635 (19 pages)

Online Publication Date: May 15, 2012

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We study the asymptotic behavior of a sequence of Dirichlet problems for second order linear operators in divergence form where the matrix $(\sigma_\varepsilon) \subset L^\infty(\Omega;{\mathbb R}^{n\times n})$ is uniformly elliptic and possibly nonsymmetric. Because of the variational principle of Cherkaev and Gibiansky [Math. Phys., 35 (1994), pp. 127–145], we are able to prove a variational characterization of the $H$-convergence of $(\sigma_\varepsilon)$ in terms of the $\Gamma$-convergence of suitably associated quadratic forms.

On the Persistence and Global Stability of Mass-Action Systems

Casian Pantea

SIAM J. Math. Anal. 44, pp. 1636-1673 (38 pages)

Online Publication Date: May 17, 2012

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This paper concerns the long-term behavior of population systems, and in particular of chemical reaction systems, modeled by deterministic mass-action kinetics. We approach two important open problems in the field of chemical reaction network theory: the Persistence Conjecture and the Global Attractor Conjecture. We study the persistence of a large class of networks called lower-endotactic and, in particular, show that in weakly reversible mass-action systems with two-dimensional stoichiometric subspace all bounded trajectories are persistent. Moreover, we use these ideas to show that the Global Attractor Conjecture is true for systems with three-dimensional stoichiometric subspace.

Global Well-Posedness of a Conservative Relaxed Cross Diffusion System

Thomas Lepoutre, Michel Pierre, and Guillaume Rolland

SIAM J. Math. Anal. 44, pp. 1674-1693 (20 pages)

Online Publication Date: May 17, 2012

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We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_i)$, where the $u_i, i=1,\ldots,I$, represent $I$ density functions, $\tilde{u}$ is a spatially regularized form of $(u_1,\ldots,u_I)$, and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous.
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