MiS Preprint Repository

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.