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Transactions of the American Mathematical Society

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Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations


Authors: Gui-Qiang Chen and Hermano Frid
Journal: Trans. Amer. Math. Soc. 353 (2001), 1103-1117
MSC (2000): Primary 35B40, 35L65; Secondary 35B35, 76N15
Published electronically: September 21, 2000
MathSciNet review: 1804414
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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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02660-X
Keywords: Compressible Euler equations, discontinuous entropy solutions, Riemann solutions, uniqueness, asymptotic stability, scaling sequence, compactness, hyperbolic conservation laws
Received by editor(s): February 8, 1999
Received by editor(s) in revised form: October 4, 1999
Published electronically: September 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society