The boundary Riemann solver coming from the real vanishing viscosity approximation

Abstract

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^ε _t + A (v^ε, εv^ε_x ) v^ε_x = εB (v^ε ) v^ε_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^ε$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.