We study conservation laws with a discontinuous flux function \({\mathfrak {f}}(\textbf{x},\lambda )\) . The flux function can be expressed as \(g(\beta (\textbf{x},\lambda ))\) , where g is locally Lipschitz, \(\beta (\textbf{x},\lambda )\) is an increasing function in \(\lambda \) for each fixed \(\textbf{x}\) , \(\nabla \beta (\cdot ,0)\) is a finite measure, and \(\beta (\cdot ,0)\) 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.
