*Published Paper*

**Inserted:** 27 apr 2003

**Last Updated:** 5 may 2011

**Journal:** International Mathematics Research Notices

**Number:** 41

**Pages:** 2205-2220

**Year:** 2003

**Abstract:**

In this paper we prove
a general existence result for bounded weak solutions
of the following class of hyperbolic systems of conservation
laws in several space dimensions:
\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) \;=\; \ov{u}_{i}(\cdot)\ ,
\end{array}
\right.
\end{equation}
where $f\in W^{1,\infty}_{loc}
$ and $\ov{u}\in L^\infty$ with $

\ov{u}

\geq c>0$
$\leb^n$-a.e. and $

\ov{u}

\in BV_{loc}$.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
*

**Keywords:**
existence, conservation laws, Hyperbolic systems, several space dimensions