数学物理学报(英文版)

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数学物理学报(英文版). 2015 (4): 761-762. DOI: 10.1016/S0252-9602(15)30019-9

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RELATIVE ENTROPY AND COMPRESSIBLE POTENTIAL FLOW

Volker ELLING

数学物理学报(英文版). 2015 (4): 763-776. DOI: 10.1016/S0252-9602(15)30020-5

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Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ, v if and only if |v| < c. For motivation some simple variations on the relative entropy theme of Dafer- mos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible per- turbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.

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BOUNDED SOLUTION OF THE RELATIVISTIC BOLTZMANN EQUATION

罗兰, 喻洪俊

数学物理学报(英文版). 2015 (4): 777-786. DOI: 10.1016/S0252-9602(15)30021-7

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In either a periodic box T^d or R^d (1≤d≤ 3), we obtain global bounded solution of the relativistic Boltzmann equation near global relativistic Maxwellian, in terms of natural mass, energy conservation and the entropy inequality.

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A STUDY ON THE BOUNDARY LAYER FOR THE PLANAR MAGNETOHYDRODYNAMICS SYSTEM

秦绪龙, 杨彤, 姚正安, 周文书

数学物理学报(英文版). 2015 (4): 787-806. DOI: 10.1016/S0252-9602(15)30022-9

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The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity μ. Under some conditions on the initial and boundary data, we show that the thickness is of the order √μ|lnμ|. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.

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TWENTY-EIGHT YEARS WITH “HYPERBOLIC CONSERVATION LAWS WITH RELAXATION”

Corrado MASCIA

数学物理学报(英文版). 2015 (4): 807-831. DOI: 10.1016/S0252-9602(15)30023-0

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This paper is a review on the results inspired by the publication "Hyperbolic conservation laws with relaxation" by Tai-Ping Liu [1],with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the (near) future.

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GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES

Debora AMADORI, Paolo BAITI, Andrea CORLI, Edda DAL SANTO

数学物理学报(英文版). 2015 (4): 832-854. DOI: 10.1016/S0252-9602(15)30024-2

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In this paper we study the problem of the global existence (in time) of weak, entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit (large) threshold, then the Cauchy problem has global solutions.

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SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

陈宇, 周忆

数学物理学报(英文版). 2015 (4): 855-875. DOI: 10.1016/S0252-9602(15)30025-4

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In this paper, we establish the existence of four families of simple wave solution for two dimensional compressible full Euler system in the self-similar plane. For the 2×2 quasilinear non-reducible hyperbolic system, there not necessarily exists any simple wave solution. We prove the result that there are simple wave solutions for this 4×4 non-reducible hyperbolic system, its simple wave flow is covered by four straight characteristics λ₀=λ₁,λ₂,λ₃ and the solutions keep constants along these lines. We also investigate the existence of simple wave solution for the isentropic relativistic hydrodynamic system in the self-similar plane.

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TIME-PERIODIC SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM

段仁军, 刘双乾

数学物理学报(英文版). 2015 (4): 876-886. DOI: 10.1016/S0252-9602(15)30026-6

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In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin's method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.

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ON THE ASYMPTOTIC DYNAMICS OF THE VLASOV-YUKAWA-BOLTZMANN SYSTEM NEAR VACUUM

Sun-Ho CHOI, Seung-Yeal HA

数学物理学报(英文版). 2015 (4): 887-905. DOI: 10.1016/S0252-9602(15)30027-8

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In this paper, we study uniform L¹-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann (V-Y-B) system. For a sufficiently small and smooth initial data, we show that classical solutions exist globally and satisfy dispersion estimates, uniform L¹-stability with respect to initial data and scattering type estimate. We show that the short range nature of interactions due to the Yukawa potential enables us to construct robust Lyapunov type functional to derive scattering states. In the zero mass limit of force carrier particles, we also show that the classical solutions to the V-Y-B system converge to that of the Vlasov-Poisson-Boltzmann (V-P-B) system in any finite time interval.

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RIGOROUS ESTIMATES ON BALANCE LAWS IN BOUNDED DOMAINS

Rinaldo M. COLOMBO, Elena ROSSI

数学物理学报(英文版). 2015 (4): 906-944. DOI: 10.1016/S0252-9602(15)30028-X

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The initial--boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in [4], striving to provide a rigorous treatment and detailed references.

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ON THE CONVERGENCE RATE OF A CLASS OF REACTION HYPERBOLIC SYSTEMS FOR AXONAL TRANSPORT

曹文涛, 黄飞敏

数学物理学报(英文版). 2015 (4): 945-954. DOI: 10.1016/S0252-9602(15)30029-1

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In this paper, we consider a class of reaction hyperbolic systems for axonal transport arising in neuroscience which can be regarded as hyperbolic systems with relaxation. We prove the BV entropy solutions of the hyperbolic systems converge toward to the unique entropy solution of the equilibrium equation at the optimal rate O(√δ)in L¹ norm as the relaxation time δ tends to zero.

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