Conservation laws with discontinuous flux functions and boundary condition (Q1397621)

The author studies the initial boundary value problem for a first order quasilinear equation \(u_t+\operatorname {div}\Phi(u)\ni f\) in some bounded domain \(\Omega\subset {\mathbb R}^N\), with zero boundary data. The flux vector \(\Phi(u)\) is supposed to have finite number of discontinuities of first type. The author introduces the notion of entropy solution and using the nonlinear semigroups theory, establishes existence and uniqueness of entropy solutions.