Article

Archive for Rational Mechanics and Analysis

, 191:1

First online:

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

  • Stefano BianchiniAffiliated withSISSA-ISAS
  • , Laura V. SpinoloAffiliated withNorthwestern University Email author 

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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.