Calculus of Variations and Geometric Measure Theory
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A. A. Contreras Hip - X. Lamy - E. Marconi

Generalized Characteristics for Finite Entropy Solutions of Burgers' Equation

created by marconi on 14 Sep 2021

[BibTeX]

Preprint

Inserted: 14 sep 2021
Last Updated: 14 sep 2021

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.


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