The boundary Riemann solver coming from the real vanishing viscosity approximation

We study the limit of the hyperbolic-parabolic approximation $$ \begin{array}{lll} v_t + \tilde{A} ( v, \, \varepsilon v_x ) v_x = \varepsilon \tilde{B}(v ) v_{xx} \qquad v \in R^N\\ \tilde \beta (v (t, \, 0)) = \bar g \\ v (0, \, x) = \bar v_0. \\ \end{array} \right. $$ The function $\tilde \beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\tilde \beta$ is not invertible. The data $\bar g$ and $\bar v_0$ are constant. When $\tilde B$ is invertible, the previous problem takes the simpler form $$ \left\{ \begin{array}{lll} v_t + \tilde{A} \big( v, \, \varepsilon v_x \big) v_x = \varepsilon \tilde{B}(v ) v_{xx} \qquad v \in \mathbb{R}^N\\ v (t, \, 0) \equiv \bar v_b \\ v (0, \, x) \equiv \bar{v}_0. \\ \end{array} \right. $$ Again, the data $\bar v_b$ and $\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, 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 $\tilde A$ can be 0. Second, as pointed out before we take into account the possibility that $\tilde B$ 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.