A note on conservation laws with discontinuous flux and L 1 initial data

We study conservation laws with a discontinuous flux function $f(x,\lambda )$ . The flux function can be expressed as $g\left(\beta \right(x,\lambda \left)\right)$ , where g is locally Lipschitz, $\beta (x,\lambda )$ is an increasing function in $\lambda$ for each fixed $x$ , $\nabla \beta (·,0)$ is a finite measure, and $\beta (·,0)$ is bounded. We consider this problem under the Audusse-Perthame entropy condition and derive a kinetic formulation. Using the kinetic approach, we prove an existence result under the assumption that the initial function belongs to $L^1$ . Uniqueness results are also presented.