Remark on the \(L_q-L_\infty\) estimate of the Stokes semigroup in a 2-dimensional exterior domain
type: Item
instance of: scholarly article
title: Remark on the \(L_q-L_\infty\) estimate of the Stokes semigroup in a 2-dimensional exterior domain
zbMATH Open document ID: 0931.35021
author: Wakako Dan
author: Yoshihiro Shibata
published in: Pacific Journal of Mathematics
publication date: 2000-02-20T00:00:00Z
review text: We show the \(L_q-L_\infty\) type estimate of the Stokes semigroup in a 2-dimensional exterior domain \(\Omega\). In our previous study, we obtained the \(L_q\)-\(L_r\) estimates of the Stokes semigroup in a 2-dimensional exterior domain, but it does not include the case that \(r=\infty\). If we go back to the representation formula of solutions to the resolvent equation, by combining several known results concerning the estimates of Stokes resolvent in \(\mathbb{R}^n\) and the asymptotic behavior of Stokes resolvent in the exterior domain near the boundary we obtain the estimate \[ \|e^{-t\mathbb{A}} {\mathbf f} \|_{L_\infty} \leq Ct^{-1/q} \|{\mathbf f} \|_{L_q} \quad \forall t>0\text{ and }{\mathbf f}\in \mathbb{J}_q(\Omega), \] where \(\mathbb{A}\) is the Stokes operator, and \(\mathbb{J}_q (\Omega)\) denotes the closure in \(L_q(\Omega) \times L_q(\Omega)\) of all solenoidal vector fields with compact support. This result can be applied to the Navier-Stokes equation to obtain the \(L_\infty\) estimate of the solution in a 2-dimensional exterior domain.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 1356077
zbMATH Keywords: representation formula of solutions
zbMATH Keywords: solenoidal vector fields
MaRDI profile type: Publication
full work available at URL: pjm.1999.189.223
OpenAlex ID: W2005329633
DOI: 10.2140/PJM.1999.189.223
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description: scientific article; zbMATH DE number 1356077