A Case study in vanishing viscosity

Stefano Bianchini, Alberto Bressan

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form ut + A(u)ux = εuxx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system ut + f(u)x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.

Original languageEnglish (US)
Pages (from-to)449-476
Number of pages28
JournalDiscrete and Continuous Dynamical Systems
Volume7
Issue number3
DOIs
StatePublished - Jul 2001

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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