Preprint 40/2003

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


 equation4

can be ill posed for suitable Lipschitz flux functions f and formula26 initial data formula28 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 formula34 and that formula36 with formula38 formula40-a.e. and formula42. 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 formula26 bounds on the distributional divergence.

23.06.2018, 02:11