TY - JOUR

T1 - On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain

AU - Enomoto, Yuko

AU - Shibata, Yoshihiro

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/6/15

Y1 - 2012/6/15

N2 - 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.

AB - 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.

KW - 2-dimensional exterior domain

KW - Compressible viscous fluid

KW - L -L decay

KW - Local energy decay

KW - Stokes semigroup

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U2 - 10.1016/j.jde.2012.03.003

DO - 10.1016/j.jde.2012.03.003

M3 - Article

AN - SCOPUS:84859716070

VL - 252

SP - 6214

EP - 6249

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -