Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions

). To begin with, we assume that the flux-function f(u) is piecewise genuinely nonlinear, in the sense that it exhibits finitely many (at most p, say) points of lack of genuine nonlinearity along each wave curve. Importantly, our analysis applies to arbitrary large p, in the sense that the constant c restricting the total variation is independent of p. Second, by an approximation argument, we prove that the existence theory above extends to general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f ^ε(u).

The main contribution in this paper is the derivation of uniform estimates for the wave curves and wave interactions (which are entirely independent of the properties of the flux-function) together with a new wave interaction potential which is decreasing in time and is a fully local functional depending upon the angle made by any two propagating discontinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p=p(v) satisfying solely the hyperbolicity condition p′(v)<0 but no convexity assumption.

Authors and Affiliations

1. Department of Mathematics Graduate School of Science and Engineering Tokyo Institute of Technology 2-12-1 Oh-okayama, Meguro-ku Tokyo 152-8551, JAPAN e-mail: iguchi@math.titech.ac.jp, , , , , , JP

   Tatsuo Iguchi

2. Centre de Mathématiques Appliquées and Centre National de la Recherche Scientifique U.M.R. 7641, Ecole Polytechnique 91128 Palaiseau, FRANCE e-mail: lefloch@cmap.polytechnique.fr, , , , , , FR

   Philippe G. LeFloch

(Accepted December 30, 2002) Published online April 23, 2003

Communicated by C. M. Dafermos

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