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Stable viscosity matrices for systems of conservation laws. (English) Zbl 0512.76067

We study a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, \(u_t + f(u)_x = 0\), \(u\in\mathbb R^m\). These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system \(u_t + f(u)_x = \nu(Du_x)_x\) as \(\nu\to 0\) if \(D\) is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation \(u_t + f'(u_0)u_x = \nu(Du_x)_{xx}\) should be well posed in \(L^2\) uniformly in \(\nu\) as \(\nu\to 0\). Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L65 Hyperbolic conservation laws
76L05 Shock waves and blast waves in fluid mechanics
76D99 Incompressible viscous fluids
76E99 Hydrodynamic stability

Citations:

Zbl 0512.76068
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References:

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