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2008

Volume 39, Issue 6, pp. 1711-2052


Structure of the Linearized Gravitational Vlasov–Poisson System Close to a Polytropic Ground State

Mohammed Lemou, Florian Méhats, and Pierre Raphaël

SIAM J. Math. Anal. 39, pp. 1711-1739 (29 pages)

Online Publication Date: February 15, 2008

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We deal in this paper with a generalized gravitational Vlasov–Poisson system that covers the three- and four-dimensional cases as well as the three-dimensional ultrarelativistic case. This system admits polytropic stationary solutions which are orbitally stable. We study in this paper the linear system obtained after a linearization close to these ground states and prove that the linearized flow displays at most algebraic instabilities. The heart of the proof is the derivation of a positivity property for the linearized Hamiltonian that implies a “quantitative” proof of the orbital stability statement. Our strategy follows the analysis by Weinstein [SIAM J. Math. Anal., 16 (1985), pp. 472–491], who obtained similar results for the nonlinear Schrödinger equation that turned out to be fundamental preliminary properties for the further description of the fine qualitative properties of the Hamiltonian system.

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

Ben Schweizer

SIAM J. Math. Anal. 39, pp. 1740-1763 (24 pages) | Cited 5 times

Online Publication Date: February 15, 2008

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We consider the one-dimensional degenerate two-phase flow equations as a model for water drive in oil recovery. The effect of oil trapping is observed in strongly heterogeneous materials with large variations in the permeabilities and in the capillary pressure curves. In such materials, a vanishing oil saturation may appear at interior interfaces and inhibit the oil recovery. We introduce a free boundary problem that separates a critical region with locally vanishing permeabilities from a strictly parabolic region and we give a rigorous derivation of the effective conservation law.

Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Dominique Blanchard, Antonio Gaudiello, and Taras A. Mel'nyk

SIAM J. Math. Anal. 39, pp. 1764-1787 (24 pages)

Online Publication Date: February 15, 2008

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We investigate the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains $\Omega_\varepsilon^+\cup\Omega^-_\varepsilon\subset{\mathbb R}^2$ depending on $\varepsilon$ in the following way. The set $\Omega_\varepsilon^+$ is a union of fine teeth, having small cross section of size $\varepsilon$ and constant height, $\varepsilon$-periodically distributed on the upper side of a horizontal thin strip with vanishing height $h_\varepsilon$, as $\varepsilon$ tends to $0^+$. The structure is clamped on the top of the teeth, with a free boundary elsewhere, and subjected to a transverse load. As $\varepsilon$ tends to $0^+$, we obtain a “continuum" bending model of rods in the limit domain of the comb, while the limit displacement is independent of the vertical variable in the rescaled (with respect to $h_\varepsilon$) strip. We show that the displacement in the strip is equal to the displacement on the base of the teeth if $h_\varepsilon\gg\varepsilon^4$. However, if the strip is thin enough (i.e., $h_\varepsilon\simeq\varepsilon^4$), we show that microscopic oscillations of the displacement in the strip, between the basis of the teeth, may produce a limit average field different from that on the base of the teeth; i.e., a discontinuity in the transmission condition may appear in the limit model.

On the Dynamics of Liquid-Vapor Phase Transition

Konstantina Trivisa

SIAM J. Math. Anal. 39, pp. 1788-1820 (33 pages) | Cited 1 time

Online Publication Date: February 15, 2008

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We consider a multidimensional model for the dynamics of liquid-vapor phase transitions. In the present context, liquid and vapor are treated as different species with different volume fractions and different molecular weights. The model presented here is a prototype of a “binary fluid mixture” and is formulated by a system that generalizes the Navier–Stokes(–Fourier) equations in Eulerian coordinates. This system takes now a new form due to the choice of rather complex constitutive relations that can accommodate appropriately the physical context. The setting of the problem presented in this work is motivated by physical considerations. The transport fluxes satisfy rather general constitutive laws, the viscosity and heat conductivity depend on the temperature, and the pressure law is a nonlinear function of the temperature depending on the mass density fraction of the vapor (liquid) in the fluid as well as the molecular weights of the individual species. The existence of globally defined weak solutions of the relevant system of partial differential equations that generalizes the Navier–Stokes(–Fourier) equations for compressible fluids is established by using weak convergence methods, and compactness and interpolation arguments in the spirit of Feireisl [Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004] and Lions [Mathematical Topics in Fluid Mechanics, Vol. 2, The Clarendon Press, Oxford University Press, New York, 1998].

Reachable and Unreachable Sets in the Scattering Problem for the Acoustical Equation in ${\mathbb{R}}^3$

Mikhail I. Belishev and Alexsei F. Vakulenko

SIAM J. Math. Anal. 39, pp. 1821-1850 (30 pages) | Cited 1 time

Online Publication Date: February 15, 2008

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The scattering problem is to find $u=u^f(x,t)$ satisfying $u_{tt} - \Delta u + qu = 0$, $(x,t) \in \mathbb{R}^3 \times (-\infty,\infty)$; $u \mid_{|x|<-t} = 0$, $t < 0$; ${\lim_{s \to \infty} su((s+\tau)\omega,-s)} = f(\tau,\omega)$, $(\tau,\omega) \in [0,\infty) \times S^2$ for a real-valued smooth compactly supported potential $q = q(x)$ and a control $f \in {\cal F} = L_2([0,\infty)$; $L_2(S^2))$. The corresponding control problem is as follows: given $y \in {\cal H} = L_2(\mathbb{R}^3)$ find $f \in {\cal F}$ provided $u^f(\cdot,0) = y$; the reachable set is ${\cal U}= \bigl\{u^f(\cdot,0) \mid f \in {\cal F}\bigr\}$; the subspace of unreachable states is ${\cal D} = {\cal H} \ominus {\cal U}$. The main subject of the paper is the structure of ${\cal U}$ and ${\cal D}$. We present an example of the finite energy solution $u^f$ satisfying $u^f|_{|x|<|t|} = 0$, i.e., vanishing simultaneously in the past and future cones (reversing wave) and we introduce the set of points at which such a “reverse effect” occurs. The existence of reversing waves turns out to be equivalent to the lack of controllability ${\cal D} \neq \{0\}$. Cauchy data of such waves belong to the classes $D_\mp$ of the incoming and outgoing data simultaneously; in other words, $D_- \cap D_+ \neq \{0\}$. Also, simple conditions on $f$ ensuring $\|u^f(\cdot,t)\|_{\cal H} \leq c \|f\|_{\cal F}$ for all $t \in (-\infty, \infty)$ are described. We plan to apply these results to the dynamical (time-domain) inverse problem, that is, determination of potential from the dynamical scattering data. The study of controllability is the first step towards solving this problem by the boundary control method.

Decay Estimates of a Tangential Derivative to the Light Cone for the Wave Equation and Their Application

Soichiro Katayama and Hideo Kubo

SIAM J. Math. Anal. 39, pp. 1851-1862 (12 pages)

Online Publication Date: February 22, 2008

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We consider wave equations in three space dimensions and obtain new weighted $L^\infty$-$L^\infty$ estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the Lorentz boost operators.

An Inverse Problem for a Nonlinear Parabolic Equation with Applications in Population Dynamics and Magnetics

Barbara Kaltenbacher and Michael V. Klibanov

SIAM J. Math. Anal. 39, pp. 1863-1889 (27 pages)

Online Publication Date: February 22, 2008

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The parabolic equation of this paper is a nonlinear one with the unknown coefficient depending on the derivative of the solution. A uniqueness result is proven by the method of Carleman estimates. The applicability of this result is illustrated for parameter identification problems in population dynamics and magnetics. For the latter application, we provide numerical results using a reconstruction method based on a multiharmonic formulation of the problem.

Low Regularity Local Well-Posedness of the Derivative Nonlinear Schrödinger Equation with Periodic Initial Data

Axel Grünrock and Sebastian Herr

SIAM J. Math. Anal. 39, pp. 1890-1920 (31 pages)

Online Publication Date: February 22, 2008

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The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data $u_0$ in the space $\widehat{H}^{s}_{r}(\mathbb{T})$, defined by the norms $\|u_0\|_{\widehat{H}^{s}_{r}(\mathbb{T})} = \|\langle\xi\rangle^s\widehat{u}_0\|_{\ell^{r'}_{\xi}}$, is shown in the parameter range $s \ge \frac{1}{2}$, $2>r>\frac{4}{3}$. The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.

Lower Semicontinuity of Quasi-convex Bulk Energies in SBV and Integral Representation in Dimension Reduction

Jean-François Babadjian

SIAM J. Math. Anal. 39, pp. 1921-1950 (30 pages)

Online Publication Date: February 29, 2008

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A result of Larsen concerning the structure of the approximate gradient of certain sequences of functions with bounded variation is used to present a short proof of Ambrosio's lower semicontinuity theorem for quasi-convex bulk energies in $SBV$. It enables us to generalize to the $SBV$ setting the decomposition lemma for scaled gradients in dimension reduction and also to show that, from the point of view of bulk energies, $SBV$ dimensional reduction problems can be reduced to analog ones in the Sobolev spaces framework.

Long Tails in the Long-Time Asymptotics of Quasi-Linear Hyperbolic-Parabolic Systems of Conservation Laws

Guillaume van Baalen, Nikola Popović, and C. Eugene Wayne

SIAM J. Math. Anal. 39, pp. 1951-1977 (27 pages)

Online Publication Date: March 18, 2008

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The long-time behavior of solutions of systems of conservation laws has been extensively studied. In particular, Liu and Zeng [Mem. Amer. Math. Soc., 125 (1997), pp. viii–120] have given a detailed exposition of the leading order asymptotics of solutions close to a constant background state. In this paper, we extend the analysis of Liu and Zeng by examining higher order terms in the asymptotics in the framework of the so-called two-dimensional p-system, though we believe that our methods and results also apply to more general systems. We give a constructive procedure for obtaining these terms, and we show that their structure is determined by the interplay of the parabolic and hyperbolic parts of the problem. In particular, we prove that the corresponding solutions develop long tails.

Large Time Behavior of Radially Symmetric Surfaces in the Mean Curvature Flow

Mitsunori Nara

SIAM J. Math. Anal. 39, pp. 1978-1995 (18 pages)

Online Publication Date: March 18, 2008

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The large time behavior of radially symmetric surfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow is studied. By studying a Cauchy problem, we deal with moving surfaces represented by entire graphs on a hyperplane. Here an initial surface is given by a function that is bounded and radially symmetric. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order $O(t^{-1/2})$ as time goes to infinity. The proof is based on the construction of the Green's function and the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary conditions on the asymptotic stability of constant functions that represent stationary hyperplanes in the mean curvature flow.

The Derrida–Lebowitz–Speer–Spohn Equation: Existence, NonUniqueness, and Decay Rates of the Solutions

Ansgar Jüngel and Daniel Matthes

SIAM J. Math. Anal. 39, pp. 1996-2015 (20 pages) | Cited 4 times

Online Publication Date: March 21, 2008

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The logarithmic fourth-order equation $\partial_t u + \frac12\sum_{i,j=1}^d\partial_{ij}^2(u\partial_{ij}^2\log u) = 0,$ called the Derrida–Lebowitz–Speer–Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnegative solutions in space dimensions $d\leq 3$ is shown. Furthermore, a family of entropy-entropy dissipation inequalities is derived in arbitrary space dimensions, and rates of the exponential decay of the weak solutions to the homogeneous steady state are estimated. The proofs are based on the algorithmic entropy construction method developed by the authors and on an exponential variable transformation. Finally, an example for nonuniqueness of the solution is provided.

A Fluid Dynamic Model for $T$-Junctions

Alessia Marigo and Benedetto Piccoli

SIAM J. Math. Anal. 39, pp. 2016-2032 (17 pages)

Online Publication Date: March 21, 2008

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Motivated by real road junctions, we consider a new fluid dynamic model for traffic flow on networks. In particular at $T$-junctions, beside some flows distribution and/or merging, there happen some interactions of cars coming from different roads and going to different destinations. After determining some rules to uniquely solve Riemann problems, we prove existence of solutions on complete networks for initial data with bounded variation (and their limits in $L^1_{loc}$).

Hopf Bifurcation From Viscous Shock Waves

Björn Sandstede and Arnd Scheel

SIAM J. Math. Anal. 39, pp. 2033-2052 (20 pages)

Online Publication Date: March 21, 2008

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Using spatial dynamics, we prove a Hopf bifurcation theorem for viscous Lax shocks in viscous conservation laws. The bifurcating viscous shocks are unique (up to time and space translation), exponentially localized in space, periodic in time, and their speed satisfies the Rankine–Hugoniot condition. We also prove an “exchange of spectral stability" result for super- and subcritical bifurcations and outline how our proofs can be extended to cover degenerate, over-, and undercompressive viscous shocks.
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