Unique Solutions to Hyperbolic Conservation Laws with a Strictly Convex Entropy

Abstract

Consider a strictly hyperbolic $n\times n$ system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation. If the system admits a strictly convex entropy, we give a short proof that every entropy weak solution taking values within the domain of the semigroup coincides with a semigroup trajectory. The result shows that the assumptions of ``Tame Variation" or ``Tame Oscillation", previously used to achieve uniqueness, can be removed in the presence of a strictly convex entropy.