A Case study in vanishing viscosity

Stefano Bianchini, Alberto Bressan

Research output: Contribution to journalArticle

11 Citations (Scopus)

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
StatePublished - Jul 2001

Fingerprint

Vanishing Viscosity
Viscosity
Hyperbolic Systems of Conservation Laws
Lyapunov Functionals
Viscosity Solutions
Total Variation
Hyperbolic Systems
Weak Solution
Shock
Entropy
Converge
Conservation
Curve
Fluxes
Interaction

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Analysis
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Bianchini, Stefano ; Bressan, Alberto. / A Case study in vanishing viscosity. In: Discrete and Continuous Dynamical Systems. 2001 ; Vol. 7, No. 3. pp. 449-476.
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A Case study in vanishing viscosity. / Bianchini, Stefano; Bressan, Alberto.

In: Discrete and Continuous Dynamical Systems, Vol. 7, No. 3, 07.2001, p. 449-476.

Research output: Contribution to journalArticle

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