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

DOI:
https://doi.org/10.1090/S0002-9947-00-02660-X

Published electronically:
September 21, 2000

MathSciNet review:
1804414

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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