Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

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:
24.04.2003
Published:
24.04.2003
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