A simple proof of -estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part II: Weak solutions

Authors:
Giovanni P. Galdi and Mads Kyed

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1313-1322

MSC (2010):
Primary 35Q30, 35B45, 76D07

DOI:
https://doi.org/10.1090/S0002-9939-2012-11640-5

Published electronically:
August 23, 2012

MathSciNet review:
3008878

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is the second of two papers in which simple proofs of -estimates of solutions to the steady-state three-dimensional Oseen and Stokes equations in a rotating frame of reference are given. In this part, estimates are established in terms of data in homogeneous Sobolev spaces of negative order.

**[Far06]**Reinhard Farwig,*An -analysis of viscous fluid flow past a rotating obstacle*, Tohoku Math. J. (2)**58**(2006), no. 1, 129-147. MR**2221796 (2007f:35226)****[FHM04]**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, 297-312. MR**2068783 (2005f:35078)****[Gal94]**Giovanni P. Galdi,*An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems*, Springer Tracts in Natural Philosophy. 38. New York: Springer-Verlag, 1994. MR**1284205 (95i:35216a)****[Gal02]**-,*On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications*, Friedlander, S. et al. (eds.), Handbook of mathematical fluid dynamics. Vol. 1. Amsterdam: Elsevier. 653-791, 2002. MR**1942470 (2003j:76024)****[Gal03]**-,*Steady flow of a Navier-Stokes fluid around a rotating obstacle*, Journal of Elasticity**71**(2003), 1-31. MR**2042672 (2005c:76030)****[GK11a]**Giovanni P. Galdi and Mads Kyed,*A simple proof of -estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions*, 2011. To appear in Proc. Amer. Math. Soc.**[GK11b]**-,*Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable*, Arch. Ration. Mech. Anal.**200**(2011), no. 1, 21-58. MR**2781585 (2012c:35324)****[His06]**Toshiaki Hishida,*estimates of weak solutions to the stationary Stokes equations around a rotating body*, J. Math. Soc. Japan**58**(2006), no. 3, 743-767. MR**2254409 (2007e:35226)****[KNP08]**Stanislav Kračmar, Šárka Nečasová, and Patrick Penel,*-approach to weak solutions of the Oseen flow around a rotating body*, Rencławowicz, Joanna et al. (eds.), Parabolic and Navier-Stokes equations. Part 1. Proceedings of the conference, Bedlewo, Poland, September 10-17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 259-276, 2008. MR**2547463 (2010d:35002)****[KNP10]**S. Kračmar, S. Nečasová, and P. Penel,*-approach of weak solutions to stationary rotating Oseen equations in exterior domains*, Q. Appl. Math.**68**(2010), no. 3, 421-437. MR**2676969 (2012b:76032)****[Lad69]**Olga A. Ladyzhenskaya,*The mathematical theory of viscous incompressible flow*, Second English edition, Gordon and Breach Science Publishers, New York, 1969. MR**0254401 (40:7610)****[Sil04]**Ana Leonor Silvestre,*On the existence of steady flows of a Navier-Stokes liquid around a moving rigid body*, Math. Methods Appl. Sci.**27**(2004), no. 12, 1399-1409. MR**2069156 (2005f:35251)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
35Q30,
35B45,
76D07

Retrieve articles in all journals with MSC (2010): 35Q30, 35B45, 76D07

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, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany

Email:
kyed@mathematik.tu-darmstadt.de

DOI:
https://doi.org/10.1090/S0002-9939-2012-11640-5

Received by editor(s):
August 9, 2011

Published electronically:
August 23, 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