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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 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.

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.