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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation


Authors: Ran Duan, Hongxia Liu and Huijiang Zhao
Journal: Trans. Amer. Math. Soc. 361 (2009), 453-493
MSC (2000): Primary 35L65, 35L60
Published electronically: August 15, 2008
MathSciNet review: 2439413
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Abstract: The expansion waves for the compressible Navier-Stokes equations have recently been shown to be nonlinear stable. The nonlinear stability results are called local stability or global stability depending on whether the $ H^1-$norm of the initial perturbation is small or not. Up to now, local stability results have been well established. However, for global stability, only partial results have been obtained. The main purpose of this paper is to study the global stability of rarefaction waves for the compressible Navier-Stokes equations. For this purpose, we introduce a positive parameter $ t_0$ in the construction of smooth approximations of the rarefaction wave solutions for the compressible Euler equations so that the quantity $ \ell=\frac{t_0}{\delta}$ ($ \delta$ denotes the strength of the rarefaction waves) is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. Then by using the energy method together with the continuation argument, we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as $ \ell\to +\infty$. Notice that the assumption that the quantity $ \ell$ can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially.


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Additional Information

Ran Duan
Affiliation: Laboratory of Nonlinear Analysis, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China

Hongxia Liu
Affiliation: Department of Mathematics, Jinan University, Guangzhou 510632, People’s Republic of China

Huijiang Zhao
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
Email: hhjjzhao@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04637-0
PII: S 0002-9947(08)04637-0
Keywords: Rarefaction waves, compressible Navier-Stokes equations, global stability, large initial perturbation.
Received by editor(s): April 9, 2007
Published electronically: August 15, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.