Uniqueness and Asymptotic Stability of Riemann Solutions
for the Compressible Euler Equations
Author: Gui-Qiang Chen and Hermano Frid
Title:
Uniqueness and Asymptotic Stability of Riemann Solutions
for the Compressible Euler Equations
Abstract
We prove the uniqueness of Riemann solutions in the class of entropy
solutions in $L^\infty\cap BV_{loc}$ for the $3\X3$ 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, by extending
the method introduced by DiPerna in \cite{Di5} for $2\times2$ systems.
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 the framework established in \cite{CF1,CF3} 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 $t>0$.
This article has appeared in:
Transaction American Mathematical Society,
vol. 353, No. 3, pages 1103-1117 (2000)
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Author Address
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu