Generalized Characteristics for Finite Entropy Solutions of Burgers' Equation

TL;DR

This paper establishes the existence of generalized characteristics for weak solutions of Burgers' equation with signed measure entropy productions, advancing understanding of non-entropic solutions and their stability.

Contribution

It introduces a framework for generalized characteristics of non-entropic solutions, removing previous technical assumptions and utilizing a Lagrangian representation for multidimensional scalar conservation laws.

Findings

Proves existence of generalized characteristics for solutions with signed measure entropy productions.

Provides a decomposition formula for entropy flux applicable to multidimensional scalar conservation laws.

Enhances stability analysis of non-entropic solutions in hydrodynamic limits.

Abstract

We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation \[ \partial_t u +\partial_x \frac{u^2}{2} =0, \] whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the $L^2$ stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution's hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.