On a degenerate scalar conservation law with general boundary condition

Abstract

We study a degenerate scalar conservation law on a bounded domain with non homogeneous boundary condition: $b(v)_t + \mbox{div } \Phi(v)=f $ on $Q:= (0,T) \times \Omega$, $v(0,\cdot)=v_0$ on $\Omega$ and $v=a$ on the boundary $(0,T) \times \partial \Omega$. The function $b$ is assumed to be continuous nondecreasing and to verify the normalization condition $b(0)=0.$ Existence and uniqueness of a renormalized entropy solution is established for any $\Phi \in C({{\mathbb R}};{{\mathbb R}}^N)$, $v_0\in L^\infty(\Omega)$, $f \in L^\infty(Q)$, and boundary data $a\in L^\infty(\Sigma).$