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- W2149201500 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." @default.
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- W2149201500 date "2008-10-10" @default.
- W2149201500 modified "2023-09-30" @default.
- W2149201500 title "The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation" @default.
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- W2149201500 doi "https://doi.org/10.1007/s00205-008-0177-6" @default.
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