Search

We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p-system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley & Sons, Inc.

A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL1-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.

In this paper, we introduce and study a general notion of multivalued solution for nonlinear hyperbolic systems, which need not to be strictly hyperbolic and in conservative form. Then we focus our attention on the system of convervation laws of fluid mechanics with constant pressure which is used in plasma physics. This system is in conservative form but not strictly hyperbolic, and is not solvable in the setting of measurable and bounded single valued functions. However we prove that multivalued solutions in the above sense can be found for this system. Moreover, these solutions are the physically meaningful solutions of the problem.

We prove a result of existence and uniqueness of entropy weak solutions for two nonstrictly hyperbolic systems, both a nonconservative system of two equations $$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}w + a(u){\partial_x}w = 0 $$ , and a conservative system of two equations $$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}v + {\partial_x}(a(u)v) = 0 $$ , where f: R → R is a given strictly convex function and \( a = \frac{d}{{du}}f \). We use the Volpert’s product ([19], see also Dal Maso — Le Floch — Murat [1]) and find entropy weak solutions u and w which have bounded variation while the solutions v are Borel measures. The equations for w and v can be viewed as linear hyperbolic equations with discontinuous coefficients.

We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ϵ ℝ . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.

Considerant une loi de conservation scalaire avec poids qui modelise l'evolution d'un gaz en geometrie axisymetrique ou dans un tuyere, on utilise une formule explicite obtenue precedemment pour preciser le comportement asymptotique de la solution faible entropique suivant le comportement de la fonction-flux et de la fonction-poids de l'equation

For nonlinear hyperbolic systems of conservation laws, the initial-boundary value problem is studied. Two formulations of boundary conditions are proposed: an entropy boundary inequality is derived thanks to the viscosity method, and a second formulation is based on the Riemann problem. These two formulations are equivalent for linear systems and scalar nonlinear equations. For nonlinear systems, the second formulation leads to well-posed problems. Nonlinear local structure is studied. The p-system and the isentropic Euler equations are detailed.

For nonlinear hyperbolic systems in nonconservation form, we consider weak solutions in the class of bounded functions of bounded variation. A generalized global entropy inequality is proposed and studied. In this mathematical framework, we solve the Riemann problem and prove, for the Cauchy problem, the consistancy of the random choice method for systems in nonconservation form. Our theory of entropy weak solutions is applied to nonconservative systems of elastodynamics and gasdynamics. In particular, we give here a nonconservation form of the system of conservation laws of gasdynamics, which is equivalent for weak solutions in BV.

This these is composed by two parts which are relatively independent to each other. In the first part an alternative theory of the gravitation, the so-called f(R) gravity, is studied. A first mathematical analysis is discussed on this theory, including the mathematical formulation of the Cauchy problem, the discussion on the choice of coupling, the mathematical formulation of the differential system. This system is four-order and highly involved. To establish the local well-posedness result, a series of transformations ans re-formulations is introduced and we finally arrived at a formulation, called the augmented conformal formulation with which we have managed to establish the local well-poseness theory.The second part is devoted to the analysis of a type of coupled wave and Klein-Gordon system. This kind of system arises naturally in many physical model, especially in the Einstein equation coupled with a real massive scalar field and the augmented conformal formulation of the f(R) gravity. The main difficulty to treat this type of system is the lack of symmetry: one of the conformal Killing vector filed of the linear wave operator, the scaling vector field S := t∂t+r∂r is not a conformal Killing vector field of the linear Klein-Gordon operator. To overpass this difficult, a new framework, called the hyperboloidal foliation method is introduced. With this framework we can encompass the wave equations and the Klein-Gordon equations in the same framework. This allowed us to establish a global well-posedness result for compactly supported, small amplitude initial data., Cette thèse est composée de deux parties qui sont relativement indépendantes l’un de l’autre. Dans la première partie,une autre théorie de la gravitation que l’on appelle la gravité de f(R), est étudiée. Une première analyse mathématique est discutée sur cette théorie, y compris la formulation mathématique du problème de Cauchy, la discussion sur le choix du couplage, et la formulation mathématique des équations différentielles. Ce système des équations différentielles est de quatrième ordre et très impliqué. Pour pouvoir établir l’existence locale, une série de transformations et reformulation et introduites. Elles nous amènent à une formulation que l’on l’appelle la formulation conforme augmenté. Avec cette formulation, l’existence locale est établie. La deuxième partie est consacrée à l’analyse d’un type de système non-linéaire composé des équations d’onde et équations de Klein-Gordon. Ce type de système apparaît naturellement dans de nombreux modèles physiques: le plus important, l’équation d’Einstein couplé avec un champ scalaire réel du massif et le système de la formulation conforme augmentée de la théorie de f(R). La difficulté principale est le manque de la symétrie: un des champs de vecteur de Killing conforme de l’opérateur d’onde, le champ de vecteur de scaling S := t∂ t +r∂ r, n’est pas un champ de vecteur de Killing conforme de l’opérateur de Klein Gordon. Pour franchir cette difficulté, un nouveau cadre, appelé la méthode de feuilletage hyperboloïdal, est introduit. Avec ce cadre, nous pouvons encadrer les équations d’onde et les équations de Klein-Gordon dans le même cadre. Cela nous permet d’établir un résultat d’existence globale pour les données initiales petites et localisées dans un compact.