Stable viscosity matrices for systems of conservation laws

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A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u1 + f(u)x = 0, uε{lunate}Rm, is studied. 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 ut + f(u)x = v(Dux)x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u1 + f′(u0) ux = vDuxx should be well posed in L2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion. © 1985.




Majda, A., & Pego, R. L. (1985). Stable viscosity matrices for systems of conservation laws. Journal of Differential Equations, 56(2), 229–262.

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