Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS 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

Abstract

In a recent paper Bressan has shown that the Cauchy problem for the system of conservation laws \begin{equation}\label{e:Cauchy} \left\{ \begin{array}{l} \partial_t u_i + \sum\limits^n_{\alpha=1} \partial_{x_\alpha} (f_\alpha (|u|) u_i) \;=\; 0\\ u_i (0, \cdot) \;=\; \bar{u}_i(\cdot)\ \end{array} \right. \end{equation} can be ill posed for suitable Lipschitz flux functions $f$ and $L^\infty$ initial data $\bar{u}$ 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 \eqref{e:Cauchy} assuming that $f\in W^{1,\infty}_{loc} $ and that $\bar{u}\in L^\infty$ with $|\bar{u}|\geq c>0$ $\mathcal{L}^n$-a.e. and $|\bar{u}|\in BV_{loc}$. 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 $L^\infty$ bounds on the distributional divergence.

Received:
Apr 24, 2003
Published:
Apr 24, 2003
MSC Codes:
35L65, 35L40, 34A12
Keywords:
hyperbolic systems, several dimensions, existence

Related publications

inJournal
2003 Repository Open Access
Luigi Ambrosio and Camillo De Lellis

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

In: International mathematics research notices, 2003 (2003) 41, pp. 2205-2220