Generalized Characteristics for Finite Entropy Solutions of Burgers' Equation

created by marconi on 14 Sep 2021
modified on 17 May 2022

[BibTeX]

Accepted Paper

Inserted: 14 sep 2021
Last Updated: 17 may 2022

Journal: Nonlinear Analysis
Year: 2021

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.