The sharp form of Oleĭnik’s entropy condition in several space variables

Abstract:

We investigate the conditions under which the Volpert-Kruzkov solution of a single conservation law in several space variables with flux $F$ will satisfy the simplified entropy condition $\operatorname {div} F’(u) \leqslant 1/t$, and when this condition guarantees uniqueness for given ${L^\infty }$ Cauchy data. We show that, when $F$ is ${C^1}$, our condition guarantees uniqueness iff $F$ is isotropic, and that, for such $F$, the Volpert-Kruzkov solution always satisfies our condition.