Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions

Luigi Ambrosio and Camillo De Lellis

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Submission date: 24. Apr. 2003 (revised version: April 2003)
Pages: 15
published in: International mathematics research notices, 2003 (2003) 41, p. 2205-2220 
DOI number (of the published article): 10.1155/S1073792803131327
Bibtex
MSC-Numbers: 35L65, 35L40, 34A12
Keywords and phrases: hyperbolic systems, several dimensions, existence
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Abstract:
In a recent paper Bressan has shown that the Cauchy problem for the system of conservation laws

can be ill posed for suitable Lipschitz flux functions f and initial data which are bounded away from 0. In the final part of his paper Bressan points out that the Cauchy problem could be well posed for BV initial data. In this paper we prove a general existence result for bounded weak solutions of e:Cauchy assuming that and that with -a.e. and . Our proof relies on recent results of the first author, which extend the Di Perna-Lions theory of ODE with discontinuous coefficients to BV vector fields satisfying natural bounds on the distributional divergence.