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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Enomoto, Yuko¹  Shibata, Yoshihiro^2,3 

[1] Shibaura Inst Technol, Dept Math Sci, Minuma Ku, Saitama 3378570, Japan

[2] Waseda Univ, Dept Math, Shinjuku Ku, Tokyo 1698555, Japan

[3] Waseda Univ, Res Inst Sci & Engn, Shinjuku Ku, Tokyo 1698555, Japan

关键词: Stokes semigroup;    Compressible viscous fluid;    2-dimensional exterior domain;    Local energy decay;    L-p-L-q decay;   

DOI  :  10.1016/j.jde.2012.03.003

来源: Elsevier

【 摘 要 】

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 lambda = 0, lambda being a resolvent parameter, while it is continuous up to lambda = 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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free