A simple proof of estimates for the steadystate 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), 573583
MSC (2010):
Primary 35Q30, 35B45, 76D07
Published electronically:
June 18, 2012
MathSciNet review:
2996962
Fulltext PDF
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Additional Information
Abstract: Consider a rigid body moving in a threedimensional NavierStokes liquid with a prescribed velocity and a nonzero 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 steadystate wholespace 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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 Reinhard Farwig, Toshiaki Hishida, and Detlef Müller, theory of a singular winding integral operator arising from fluid dynamics, Pac. J. Math. 215 (2004), no. 2, 297312. MR 2068783 (2005f:35078)
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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, D64289 Darmstadt, Germany
Email:
kyed@mathematik.tudarmstadt.de
DOI:
http://dx.doi.org/10.1090/S000299392012116387
Received by editor(s):
July 1, 2011
Published electronically:
June 18, 2012
Additional Notes:
The first author was partially supported by NSF grant DMS1062381
The second author was supported by the DFG and JSPS as a member of the International Research Training Group DarmstadtTokyo IRTG 1529.
Communicated by:
Walter Craig
Article copyright:
© Copyright 2012
American Mathematical Society
