Vanishing viscosity solutions for conservation laws with regulated flux

Alberto Bressan, Graziano Guerra, Wen Shen

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper we introduce a concept of “regulated function” v(t,x) of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form ut+F(v(t,x),u)x=0, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation. As an application, we obtain the existence and uniqueness of solutions for a class of 2×2 triangular systems of conservation laws with hyperbolic degeneracy.

Original languageEnglish (US)
Pages (from-to)312-351
Number of pages40
JournalJournal of Differential Equations
Volume266
Issue number1
DOIs
StatePublished - Jan 5 2019

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Triangular Systems
Vanishing Viscosity
Systems of Conservation Laws
Discontinuous Functions
Scalar Conservation Laws
Hamilton-Jacobi Equation
Viscosity Solutions
Existence and Uniqueness of Solutions
Degeneracy
Conservation Laws
Parabolic Equation
Conservation
Viscosity
Uniqueness
Fluxes
Zero
Term
Estimate
Concepts
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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Vanishing viscosity solutions for conservation laws with regulated flux. / Bressan, Alberto; Guerra, Graziano; Shen, Wen.

In: Journal of Differential Equations, Vol. 266, No. 1, 05.01.2019, p. 312-351.

Research output: Contribution to journalArticle

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