Uniqueness and Weak-BV Stability for 2×2 Conservation Laws View Full Text


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Article Info

DATE

2022-09-07

AUTHORS

Geng Chen, Sam G. Krupa, Alexis F. Vasseur

ABSTRACT

Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small BV functions which are global solutions of this equation. For any small BV initial data, such global solutions are known to exist. Moreover, they are known to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even BV) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-BV uniqueness result, where only one of the solutions is supposed to be small BV, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with 2 unknowns. The method is L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} based, and builds up from the theory of a-contraction with shifts, where suitable weight functions a are generated via the front tracking method. More... »

PAGES

299-332

References to SciGraph publications

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  • 1994-07. Kinetic formulation of the isentropic gas dynamics andp-systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1983-03. Convergence of the viscosity method for isentropic gas dynamics in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2011-05-15. Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2010-07-07. L2 Stability Estimates for Shock Solutions of Scalar Conservation Laws Using the Relative Entropy Method in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1979-06. The second law of thermodynamics and stability in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2020-10-15. Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier–Stokes systems in INVENTIONES MATHEMATICAE
  • 1999-10. L1 Stability Estimates for n×n Conservation Laws in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2001-11. Strong Traces for Solutions of Multidimensional Scalar Conservation Laws in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2003-07-21. Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1983. Shock Waves and Reaction—Diffusion Equations in NONE
  • 1997-12. Uniqueness of Weak Solutions to Systems of Conservation Laws in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1994-05. Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy in MATHEMATICAL NOTES
  • 2007-06-01. Strong Traces for Solutions to Scalar Conservation Laws with General Flux in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2002-06. Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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    http://scigraph.springernature.com/pub.10.1007/s00205-022-01813-0

    DOI

    http://dx.doi.org/10.1007/s00205-022-01813-0

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