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

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 div F'(u) s l/t, and when this condition guarantees uniqueness for given LX Cauchy data. We show that, when F is C1, our condition guarantees uniqueness iff F is isotropic, and that, for such F, the Volpert-Kruzkov solution always satisfies our condition.