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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Local Existence of Classical Solutions to Shallow Water Equations with Cauchy Data Containing Vacuum SIAM J. Math. Anal. 44, pp. 541-567 (27 pages) Online Publication Date: March 08, 2012
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In this paper, we investigate the Cauchy problem for the rotating shallow water equations with physical viscosity. We obtain the local existence of classical solutions without assuming the initial height is small or a small perturbation of some constant status. Moreover, the initial vacuum is allowed and the spatial measure of the set of vacuum can be arbitrarily large. In particular, the initial height can even have compact support; in this case, a blow-up example is given. |
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The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle SIAM J. Math. Anal. 44, pp. 568-602 (35 pages) Online Publication Date: March 08, 2012
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In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. Most results except the ones in sections 3 and 6 concern radial solutions. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions. |
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Asymptotic Stability of Boundary Layers to the Euler–Poisson Equations Arising in Plasma Physics SIAM J. Math. Anal. 44, pp. 761-790 (30 pages) Online Publication Date: March 22, 2012
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The main concern of the present paper is to analyze the behavior of a boundary layer, called a sheath, which appears over a material in contact with a plasma. The well-known Bohm criterion claims the velocity of positive ions should be faster than a certain constant for the formation of a sheath. The behavior of positive ions is governed by the Euler–Poisson equations. Mathematically, the sheath is understood as a monotone stationary solution, whose existence and asymptotic stability in one-dimensional space were proved in Suzuki's previous work. However the stability was proved under the assumption stronger than the Bohm criterion. In the present paper, we refine these results by proving the stability theorem exactly under the Bohm criterion in the spatial dimension up to three. We also deal with the degenerate case in which the Bohm criterion is marginally fulfilled. |
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Homogenization and Two-Scale Convergence SIAM J. Math. Anal. 23, pp. 1482-1518 (37 pages) Online Publication Date: August 01, 2006
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Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations. |
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Characterization of Radially Symmetric Finite Time Blowup in Multidimensional Aggregation Equations SIAM J. Math. Anal. 44, pp. 651-681 (31 pages) Online Publication Date: March 08, 2012
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This paper studies the transport of a mass $\mu$ in $\mathbb{R}^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent $p$ we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent. |
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A Variational Approach to the Stationary Solutions of the Burgers Equation SIAM J. Math. Anal. 44, pp. 682-698 (17 pages) Online Publication Date: March 13, 2012
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We consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini et al. [Comm. Pure Appl. Math., 64 (2011), pp. 649–696], we analyze a Lyapunov functional for such an equation which gives the large deviations asymptotic of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. In particular, we focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has a one-parameter family of minimizers and we compute the sharp asymptotic cost corresponding to a given shift of the stationary solution. |
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SIAM J. Math. Anal. 44, pp. 633-650 (18 pages) Online Publication Date: March 08, 2012
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We prove the local-in-time existence of smooth solutions of the Navier–Stokes equations of compressible, barotropic flow posed in a bounded set in $\mathbb{R}^3$ with a mixed boundary condition sufficiently general to include the condition proposed by Navier that on the boundary of the spatial domain the velocity should be proportional to the projection of the stress onto the tangent plane. |
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Existence of Weak Solutions for a Class of Semilinear Stochastic Wave Equations SIAM J. Math. Anal. 44, pp. 906-925 (20 pages) Online Publication Date: April 03, 2012
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We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of $\mathbb{R}^d$ driven by a possibly discontinuous square integrable martingale. |
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The Variational Formulation of the Fokker--Planck Equation SIAM J. Math. Anal. 29, pp. 1-17 (17 pages) Online Publication Date: August 01, 2006
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The Fokker--Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker--Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker--Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker--Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric. |
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Boundary Effects in the Gradient Theory of Phase Transitions SIAM J. Math. Anal. 44, pp. 926-945 (20 pages) Online Publication Date: April 03, 2012
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We consider the van der Waals' free energy functional, with scaling parameter $\varepsilon$, in the plane domain $\mathbb{R}_+\times \mathbb{R}_+$, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries $\mathbb{R}_+ \times\{0\}$ and $\mathbb{R}_+\times\{+\infty\}$, and free boundary conditions on $\{+\infty\}\times\mathbb{R}_+$. Finally, the datum on $\{0\}\times \mathbb{R}_+$ is chosen in such a way that the interface between the pure phases is pinned at some point $(0,y)$. We show that there exists a critical scaling, $y=y_\varepsilon$, such that, as $\varepsilon\to 0$, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations. |
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Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$ SIAM J. Math. Anal. 44, pp. 1077-1102 (26 pages) Online Publication Date: April 12, 2012
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This paper deals with a degenerate diffusion Patlak–Keller–Segel system in $n\geq 3$ dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent $m=2n/(n+2)$, which is smaller than the usual exponent $m^{*}=2-2/n$ used in other studies. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant, and there is a family of stationary solutions $U_{\lambda,x_0}(x)=C(\frac{\lambda} {\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$ $\forall \lambda>0$, $x_0\in {\mathbb R}^n$. For radially symmetric solutions, we prove that if the initial data are strictly below $U_{\lambda,0}(x)$ for some $\lambda$, then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data are strictly above $U_{\lambda,0}(x)$ for some $\lambda$, then the solution either blows up at a finite time or has a mass concentration at $r=0$ as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the $L^m$ norm for initial data is larger than the $L^m$ norm of $U_{\lambda,x_0}(x)$, which is constant independent of $\lambda$ and $x_0$, and the free energy of initial data is smaller than that of $U_{\lambda,x_0}(x)$. |
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Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media SIAM J. Math. Anal. 42, pp. 125-144 (20 pages) Online Publication Date: February 17, 2010
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We study the homogenization problem for a convection-diffusion equation in a periodic porous medium in the presence of a chemical reaction on the pores' surface. Mathematically this model is described in terms of a solution to a system of convection-diffusion equations in the medium and ordinary differential equation defined on the pores' surface. These equations are coupled through the boundary condition for the convection-diffusion problem. Under an appropriate choice of scaling factors (large Péclet and Damköhler numbers), we obtain the homogenized problem in a moving frame whose effective velocity does actually depend on the chemical reaction. |
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Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations SIAM J. Math. Anal. 44, pp. 603-632 (30 pages) Online Publication Date: March 08, 2012
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We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Lévy processes (e.g., the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Lévy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover and extend to new equations a large part of the known error estimates in the literature. |
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The Periodic Unfolding Method in Domains with Holes SIAM J. Math. Anal. 44, pp. 718-760 (43 pages) Online Publication Date: March 20, 2012
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We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with nonhomogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding. |
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SIAM J. Math. Anal. 44, pp. 993-1018 (26 pages) Online Publication Date: April 10, 2012
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This article studies the solutions in $L^2$ of a steady transport equation with a divergence-free driving velocity that is $H^1$ in a Lipschitz domain of $\mathbb{R}^d$. Since the velocity is assumed fully nonhomogeneous on the boundary, existence and uniqueness of solution require a boundary condition. A new Green's formula allows us to define the normal component of $z\mathbf{u}$ on the boundary, where $z$ denotes the stress and $\mathbf{u}$ the velocity. A substantial part of the article is devoted to properties of a truncature operator in the space where $z$ and $\mathbf{u}\,.\,\nabla z$ are $L^2$. By means of these properties, which allow us to prove density results, and by using in addition a nonbounded linear operator from $L^2$ to $L^2$, we establish existence and uniqueness of the solution for the transport equation with a boundary condition on the open part where the normal component of $\mathbf{u}$ is strictly negative. |
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Radially Symmetric Solutions of a Tempered Diffusion Equation. A Porous Media, Flux-Limited Case SIAM J. Math. Anal. 44, pp. 1019-1049 (31 pages) Online Publication Date: April 10, 2012
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In this paper we provide analytical and/or numerical evidences of the qualitative properties of discontinuity fronts, developed by the solutions of the relativistic heat equation (and some porous media variants). We study the local-in-time existence of radially symmetric smooth solutions (inside the support) for smooth initial conditions whose only discontinuities are at the boundary of its support. Then we show some numerical experiments that permit us to conjecture the regularity and qualitative properties of entropy solutions of the relativistic heat equation and some porous media–type variants. |
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Thermodynamics of Rate-independent Processes in Viscous Solids at Small Strains SIAM J. Math. Anal. 42, pp. 256-297 (42 pages) Online Publication Date: March 05, 2010
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So-called generalized standard solids (of the Halphen–Nguyen type) involving also activated rate-independent processes such as plasticity, damage, or phase transformations are described as a system of a momentum equilibrium equation and a variational inequality for inelastic evolution of internal-parameter variables. The stored energy is considered as temperature dependent and then the thermodynamically consistent system is completed with the heat-transfer equation. Existence of a suitably defined “energetic” solution is proved by a nontrivial combination of theory of rate-independent processes by Mielke et al. [Handbook of Differential Equations, Elsevier, Amsterdam, 2005, pp. 461–559; Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean, and R. Farwig, eds., Shaker Ver., Aachen, 1999, pp. 117–129; Nonlinear Differ. Equ. Appl., 11 (2004), pp. 151–189; Arch. Ration. Mech. Anal., 162 (2002), pp. 137–177] adapted for coupling with viscous/inertial effects and of sophisticated estimates by Boccardo and Gallouët of the temperature gradient of the heat equation with $L^1$-data. Illustrative examples are presented, too. |
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SIAM J. Math. Anal. 44, pp. 966-992 (27 pages) Online Publication Date: April 10, 2012
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We consider the system of equations governing an incompressible immiscible two-phase flow within a heterogeneous porous medium made of two different rock types. Since the capillary pressure function depends on the rock type, the capillary pressure field might be discontinuous at the interface between the rocks. We introduce multivalued phase pressures to give a sense to the transmission conditions at the interface. We prove the existence of a solution for such a flow by passing to the limit in regularizations of the problem. |
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Singular Limits and Convergence Rates of Compressible Euler and Rotating Shallow Water Equations SIAM J. Math. Anal. 44, pp. 1050-1076 (27 pages) Online Publication Date: April 10, 2012
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With solid-wall boundary condition and ill-prepared initial data, we prove the singular limits and convergence rates of compressible Euler and rotating shallow water equations towards their incompressible counterparts. A major issue is that fast acoustic waves contribute to the slow vortical dynamics at order one and do not damp in any strong sense. Upon averaging in time, however, such a contribution vanishes at the order of the singular parameters (i.e., Mach/Froude/Rossby numbers). In particular, convergence rates of the compressible dynamics, when projected onto the slow manifold, are estimated explicitly in terms of the singular parameters and Sobolev norms of the initial data. The structural condition of a vorticity equation plays a key role in such an estimation as well as in proving singular-parameter-independent life spans of classical solutions. |
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Wright–Fisher Diffusion in One Dimension SIAM J. Math. Anal. 42, pp. 568-608 (41 pages) Online Publication Date: March 19, 2010
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We analyze the diffusion processes associated to equations of Wright–Fisher type in one spatial dimension. These are associated to the degenerate heat equation $\partial_{t}u=a(x)\partial_{x}^{2}u+b(x)\partial_{x}u$ on the interval $[0,1]$, where $a(x)>0$ on the interior and vanishes simply at the end points and $b(x)\partial_{x}$ is a vector field which is inward pointing at both ends. We consider various aspects of this problem, motivated by their applications in biology, including a sharp regularity theory for the “zero flux” boundary conditions, as well as an analysis of the infinitesimal generators of the $\mathcal{C}^{m}$-semigroups and their adjoints. Using these results we obtain precise asymptotics of solutions of this equation, both as $t\to0,\infty$ and as $x\to0,1$. |
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