On uniqueness of solutions to conservation laws verifying a single entropy condition

Abstract

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.