On Uniqueness of Solutions to Conservation Laws Verifying a Single Entropy Condition

TL;DR

This paper proves the uniqueness of solutions for scalar conservation laws with a single entropy without relying on Hamilton-Jacobi connections, introducing new techniques that could extend to systems.

Contribution

It establishes the single entropy uniqueness result for scalar conservation laws without Hamilton-Jacobi reliance, offering new methods potentially applicable to systems.

Findings

Proves uniqueness for scalar conservation laws with one entropy without Hamilton-Jacobi

Introduces new techniques for analyzing conservation law solutions

Provides a foundation for extending results to systems of conservation laws

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 Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math., 62(4):687--700, 2004]. 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.