Stable Viscosity Matrices for Systems of Conservation Laws

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, II, + f(u), =0, u E R”, 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 u, + f(u), = v(Du,), as v -t 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u, + f ‘(a,,) a, = vDu,, 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.