Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws

Dechun Tan, Tong Zhang, Tung Chang, Yuxi Zheng

Research output: Contribution to journalArticle

184 Scopus citations

Abstract

For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are w*-limits in L1 of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated.

Original languageEnglish (US)
Pages (from-to)1-32
Number of pages32
JournalJournal of Differential Equations
Volume112
Issue number1
DOIs
StatePublished - Aug 1994

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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