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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
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by Gui-Qiang Chen and Hermano Frid PDF
Trans. Amer. Math. Soc. 353 (2001), 1103-1117 Request permission

Abstract:

We prove the uniqueness of Riemann solutions in the class of entropy solutions in $L^\infty \cap BV_{loc}$ for the $3\times 3$ system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global $L^2$-stability of the Riemann solutions even in the class of entropy solutions in $L^\infty$ with arbitrarily large oscillation for the $3\times 3$ system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under $L^1$ perturbation of the Riemann initial data, as long as the corresponding solutions are in $L^\infty$ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions $U(x,t)$, piecewise Lipschitz in $x$, for any $t>0$.
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Additional Information
  • Gui-Qiang Chen
  • Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208
  • MR Author ID: 249262
  • ORCID: 0000-0001-5146-3839
  • Email: gqchen@math.northwestern.edu
  • Hermano Frid
  • Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil
  • Email: hermano@im.ufrj.br
  • Received by editor(s): February 8, 1999
  • Received by editor(s) in revised form: October 4, 1999
  • Published electronically: September 21, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 1103-1117
  • MSC (2000): Primary 35B40, 35L65; Secondary 35B35, 76N15
  • DOI: https://doi.org/10.1090/S0002-9947-00-02660-X
  • MathSciNet review: 1804414