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arXiv · 1805.01766

Vanishing Viscosity Solutions for Conservation Laws with Regulated Flux

Alberto Bressan·Graziano Guerra·Wen Shen
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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 $2\times2$ triangular systems of conservation laws with hyperbolic degeneracy.

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Alberto Bressan, Graziano Guerra, Wen Shen. 2018-05-04. Vanishing Viscosity Solutions for Conservation Laws with Regulated Flux. https://arxiv.org/abs/1805.01766

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