On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain

Yuko Enomoto, Yoshihiro Shibata

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Abstract

In this paper, we proved the local energy decay and some L p-L q decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in ℝ 2 has a logarithmical singularity at λ=0, λ being a resolvent parameter, while it is continuous up to λ=0 in ℝ 3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L p-L q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L p-L q estimates in ℝ 2 by a cut-off technique.

Original languageEnglish
Pages (from-to)6214-6249
Number of pages36
JournalJournal of Differential Equations
Volume252
Issue number12
DOIs
Publication statusPublished - 2012 Jun 15

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Keywords

  • 2-dimensional exterior domain
  • Compressible viscous fluid
  • L -L decay
  • Local energy decay
  • Stokes semigroup

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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