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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation
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by Ran Duan, Hongxia Liu and Huijiang Zhao PDF
Trans. Amer. Math. Soc. 361 (2009), 453-493 Request permission

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
  • Received by editor(s): April 9, 2007
  • Published electronically: August 15, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 453-493
  • MSC (2000): Primary 35L65, 35L60
  • DOI: https://doi.org/10.1090/S0002-9947-08-04637-0
  • MathSciNet review: 2439413