Preprint 1999-045

A Note on Front Tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations and Entropy Solutions of Scalar Conservation Laws

Kenneth Hvistendahl Karlsen and Nils Henrik Risebro

Abstract: We give a direct proof of the well known equivalence between the Kruzkov-Volpert entropy solution for the scalar conservation law $p_t + H(p)_x=0$ and the Crandall-Lions viscosity solution of the Hamilton-Jacobi equation $u_t + H(u_x)=0$. In our proof we work directly with the defining entropy and viscosity inequalities and do not, as is usually done, exploit the convergence of the viscosity method. The proof is based on establishing the equivalence directly for a "dense" set of flux functions $H$ and initial data $p_0$/$u_0$. In the course of doing so, we translate front tracking for scalar conservation laws to Hamilton-Jacobi equations and derive some of its properties.

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Kenneth Hvistendahl Karlsen, <>
Nils Henrik Risebro, <>
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<> December 27 1999.

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