Generalized characteristics for finite entropy solutions of Burgers' equation. (June 2022)

• Record Type: Journal Article
• Authors: Contreras Hip, Andres A.; Lamy, Xavier; Marconi, Elio
• Is Part Of: Nonlinear analysis. Volume 219(2022)
• Publication Date: 2022-06
• Subjects: 35L60; Generalized characteristics -- Finite entropy solutions -- Burgers' equation -- Lagrangian representation -- L2 stability of shocks; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112804
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation ∂ t u + ∂ x 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.