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 language | English |
|---|---|
| Pages (from-to) | 6214-6249 |
| Number of pages | 36 |
| Journal | Journal of Differential Equations |
| Volume | 252 |
| Issue number | 12 |
| DOIs | |
| Publication status | Published - 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
Cite this
On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain. / Enomoto, Yuko; Shibata, Yoshihiro.
In: Journal of Differential Equations, Vol. 252, No. 12, 15.06.2012, p. 6214-6249.Research output: Contribution to journal › Article
}
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
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
UR - http://www.scopus.com/inward/record.url?scp=84859716070&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84859716070&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2012.03.003
DO - 10.1016/j.jde.2012.03.003
M3 - Article
VL - 252
SP - 6214
EP - 6249
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 12
ER -