Viscosity Solutions for Hyperbolic Systems where Shock Curves are Straight Lines

Abstract

AbstractConsider 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) KeywordsCauchy ProblemViscosity SolutionHyperbolic SystemContinuous SemigroupSmooth Initial DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.