Abstract:We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in $L^1$, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form \cite{KarlsenTowersRisebro:stable}, we show that the solution operator is $L^1$ contractive. These results generalize the corresponding results of Kruzhkov and Towers et al.

**Paper:**- Available as PDF (3.3 Mbytes).
**Author(s):**- Giuseppe Maria Coclite, <coclite@sissa.it>
- Nils Henrik Risebro, <nilshr@math.uio.no>
**Publishing information:****Comments:****Submitted by:**- <nilshr@math.uio.no> December 18 2002.

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