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The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Fundamental Research 2024年第2期4卷 270-283页

作者： Shijie Dong philippe g.lefloch Zhen Lei Southern University of Science and Technology SUSTech International Center for Mathematicsand Department of MathematicsShenzhen 518055China Fudan University School of Mathematical Sciences220 Handan RoadShanghai 200433China Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique Sorbonne Université4 Place JussieuParis 75252France Shanghai Center for Mathematical Sciences Shanghai 200433China

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions,provided that the initial data are compactly supported and sufficiently small in Sobolev *** this work,Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions.A natural question then arises whether the time-growth is a true phenomenon,despite the possible conservation of basic *** the present paper,we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

关键词： Quasilinear wave equation global-in-time solution Uniform energy bounds Quadratic null nonlinearity Hyperboloidal foliation method Vector field method

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A geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime

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Communications in Computational Physics 2014年第3期15卷 827-852页

作者： philippe g.lefloch Hasan Makhlof Laboratoire Jacques-Louis Lions&Centre National de la Recherche Scientifique Université Pierre et Marie Curie(Paris 6)4 Place Jussieu75252 ParisFrance

We consider the relativistic Euler equations governing spherically symmetric,perfect fluid flows on the outer domain of communication of Schwarzschild spacetime,and we introduce a version of the finite volume method which is formulated from the geometric formulation(and thus takes the geometry into account at the discretization level)and is well-balanced,in the sense that it preserves steady solutions to the Euler equations on the curved geometry under *** order to formulate our method,we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild ***,we describe a geometry-preserving,finite volume method which is based from the family of steady solutions to the Euler *** scheme is second-order accurate and,as required,preserves the family of steady solutions at the discrete *** experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave *** an application,we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution,taking the overall effect of the perturbation into account.

关键词： Compressible fluid Euler system Schwarzschild spacetime geometry-preserving finite volume method steady state

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Hyperbolic Conservation Laws on *** Error Estimate for Finite Volume Schemes

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Acta Mathematica Sinica,English Series 2009年第7期25卷 1041-1066页

作者： philippe g.lefloch Baver OKUTMUSTUR Wladimir NEVES Laboratoire Jacques-Louis Lions Centre National de la Recherche ScientifiqueUniversité de Paris 6 Instituto de Matemática Universidade Federal do Rio de Janeiro C.P. 68530Cidade Universitária 21945-970

Following Ben-Artzi and lefloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and lefloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

关键词： Hyperbolic conservation law entropy solution finite volume scheme error estimate discrete entropy inequality convergence rate

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KINETIC FUNCTIONS IN MAgNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT

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Acta Mathematica Scientia 2009年第6期29卷 1684-1702页

作者： philippe g. Le Siddhartha Mishra FlochLaboratoire Jacques-Louis Lions Centre National de la Recherche ScientifiqueUniversit Pierre et Marie Curie (Paris 6) 4 Place Jussieu 75258 Paris France Seminar for Applied Mathematics D-Math ETH HG Raemistrasse Zürich-8092 Switzerland

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle competition that takes place between the hyperbolic, diffusive, and dispersive parts of the system. Following Abeyratne, Knowles, lefloch, and Truskinovsky, who studied similar questions arising in fluid and solid flows, we determine the associated kinetic function which characterizes the dynamics of undereompressive shocks driven by resistivity and Hall effect. To this end, we design a new class of ＂schemes with eontroled dissipation＂, following recent work by lefloch and Mohammadian. It is now recognized that the equivalent equation associated with a scheme provides a guideline to design schemes that capture physically relevant, nonclassical shocks. We propose a new class of schemes based on high-order entropy conservative, finite differences for the hyperbolic flux, and high-order central differences for the resistivity and Hall terms. These schemes are tested for several regimes of （co-planar or not） initial data and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic functions in magnetohydrodynamics.

关键词： hyperbolic conservation law magnetohydrodynamics magnetic resistivity Hall effect undercompressive shock kinetic function

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