Vanishing Viscosity Solutions for Conservation Laws with Regulated Flux

2018/05/04 by Bressan, Alberto, Guerra, Graziano, Shen, Wen · 2 citations
#35L65 #35R05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.01766

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 u_t+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.

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• Scalar Approach to ARZ-Type Systems of Conservation Laws
• A convergence rate result for front tracking approximations of conservation laws with discontinuous flux

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