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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