Viscosity Solutions for Hyperbolic Systems where Shock Curves are Straight Lines

Abstract

Consider a strictly hyperbolic n x n system of conservation laws in one space dimension:

$${{u}_{t}} + f{{(u)}_{x}} = 0.$$

(1)

Assuming that the initial data has small total variation, the global existence of weak solutions was proved by Glimm [9], while the uniqueness and stability of entropy admissible BV solutions was recently established in a series of papers [2 4 5 6 7 13]. See also [3] for a comprehensive presentation of these results. A long standing open question is whether these discontinuous solutions can be obtained as vanishing viscosity limits. More precisely, given a smooth initial data $\bar{u}:\mathbb{R} \mapsto {{\mathbb{R}}^{n}}$ with small total variation, consider the parabolic Cauchy problem

$${{u}_{t}} + A(u){{u}_{x}} = \varepsilon {{u}_{{xx}}},$$

(2)

$$u(0,x) = \bar{u}(x).$$

(3)