The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

Abstract

We study the limit of the hyperbolic–parabolic approximation

$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A} \left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx} \quad v^{\varepsilon} \in \mathbb{R}^N \cr \tilde{\rm \ss}(v^{\varepsilon} (t, \, 0)) \equiv \bar g \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$

The function ${\tilde {\ss}}$ is defined in such a way as to guarantee that the initial boundary value problem is well posed even if ${\tilde {B}}$ 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

$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A}\left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx}\quad v^{\varepsilon} \in \mathbb{R}^N \cr v^{\varepsilon} (t, \, 0) \equiv \bar v_b \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$

Again, the data ${\bar {v}_b}$ and ${\bar {v}_0}$ are constant. The conservative case is included in the previous formulations. Convergence of the ${v^{\varepsilon}}$ , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, 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 this condition is not satisfied, then pathological behaviors may occur.