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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.

This paper treats the system of conservation laws of nonlinear elastodynamics in the form recently reviewed by B. J. Plohr and D. H. Sharp (Adv. Appl. Math.9, 481–499 (1989)). Following X. Garaizar (Acta Appl. Math., to appear), we consider hyperelastic materials whose specific internal energy function is the sum of an isotropic part and a small anisotropic one. For this system of equations, we present a second order accurate Godunov-type finite difference scheme. Our scheme is based in particular on an approximation of the solution of a Riemann problem obtained by using only shock waves. Several computational results are presented.

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.