A simple proof of -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

MathSciNet review:
2996962

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Abstract: Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity and a non-zero angular velocity that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen ( ) or Stokes () 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 . 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 and . In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space .

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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

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