Viscosity solutions and standard Riemann semigroup for conservation laws with boundary
The authors consider two different formulations of the initial boundary value problem (IBVP) refered as characteristic and noncharacteristic for the following conservation law system: \(u_t+ [F(u)]_x =0\), where \(F\) is smooth and each characteristic field in \(DF\) is either linearly degenerate or genuinely nonlinear. Two ``standard Riemann semigroups'' (SRS) generated by the problem under consideration are defined here. It is shown that if an SRS exists, then it is unique, and its trajectories yield solutions to the IBVP in each of the following two cases: characteristic and noncharacteristic IBVP. Moreover, the trajectories are viscosity solutions of the IBVP for the considered system. Conversely, a viscosity solution with small total variation coincides with the corresponding semigroup trajectory as soon as SRS exists.
characteristic and noncharacteristic formulation
viscosity solution with small total variation
scientific article; zbMATH DE number 1183140