A review of recent applications of the relative entropy method to discontinuous solutions of conservation laws

Vasseur, Alexis

doi:10.1090/qam/1667

Author: Alexis F. Vasseur
Journal: Quart. Appl. Math. 81 (2023), 553-565
MSC (2020): Primary 35L65; Secondary 76N15, 35L45, 35B35
DOI: https://doi.org/10.1090/qam/1667
Published electronically: April 26, 2023
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Abstract: Dafermos [Arch. Rational Mech. Anal. 70 (1979), pp. 167–179] proved the weak/strong principle for conservation laws. It states that Lipschitz solutions to conservation laws endowed with convex entropies are unique and stable among weak solutions. The method, based on relative entropy, was extended by Di Perna [Indiana Univ. Math. J. 28 (1979), pp. 137–188] to show the uniqueness of shocks among weak solutions with strong traces. This theory has been recently revisited with the notion of weighted contractions up to shifts. We review in this paper recent applications of this method, including the weak/BV principle and the stability of discontinuous solutions among inviscid double limits of Navier-Stokes systems.

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