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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simple proof of $ L^{q}$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions


Authors: Giovanni P. Galdi and Mads Kyed
Journal: Proc. Amer. Math. Soc. 141 (2013), 573-583
MSC (2010): Primary 35Q30, 35B45, 76D07
Published electronically: June 18, 2012
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Abstract: Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity $ \xi \in \mathbb{R}^3$ and a non-zero angular velocity $ \omega \in \mathbb{R}^3\setminus \{ 0\}$ that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen ( $ \xi \neq 0$) or Stokes ($ \xi =0$) equations in a rotating frame of reference. We will consider the corresponding steady-state whole-space problem. Our main result in this first part concerns elliptic estimates of the solutions in terms of data in $ L^{q}(\mathbb{R}^3)$. Such estimates have been established by R. Farwig in Tohoku Math. J., Vol. 58, 2006, for the Oseen case, and R. Farwig, T. Hishida, and D. Müller in Pacific J. Math., Vol. 215 (2), 2004, for the Stokes case. We introduce a new approach resulting in an elementary proof of these estimates. Moreover, our method yields more details on how the constants in the estimates depend on $ \xi $ and $ \omega $. In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space $ D^{-1,q}_0(\mathbb{R}^3)$.


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Additional Information

Giovanni P. Galdi
Affiliation: Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
Email: galdi@pitt.edu

Mads Kyed
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany
Email: kyed@mathematik.tu-darmstadt.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11638-7
PII: S 0002-9939(2012)11638-7
Received by editor(s): July 1, 2011
Published electronically: June 18, 2012
Additional Notes: The first author was partially supported by NSF grant DMS-1062381
The second author was supported by the DFG and JSPS as a member of the International Research Training Group Darmstadt-Tokyo IRTG 1529.
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society