Stability of Degenerate Stationary Waves for Viscous Gases

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t-α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2α=L2(ℝ+;(1+x)αdx). This convergence rate t-α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L2α for α > α *(q) with another critical value α*(q). Our stability analysis is based on the space-time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.

Original languageEnglish
Pages (from-to)735-762
Number of pages28
JournalArchive for Rational Mechanics and Analysis
Volume198
Issue number3
DOIs
Publication statusPublished - 2010 Sep 17

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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