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 F will satisfy the simplified entropy condition div F ′ ( u ) ⩽<!-- ⩽ --> 1 / t div\,F’(u) 1/t , and when this condition guarantees uniqueness for given L ∞<!-- ∞ --> {L^∞ } Cauchy data. We show that, when F F is C 1 {C^1} , our condition guarantees uniqueness iff F F is isotropic, and that, for such F F , the Volpert-Kruzkov solution always satisfies our condition.