Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


A simple proof of $ L^{q}$-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
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 $ L^{q}$-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.

References [Enhancements On Off] (What's this?)

  • [Far06] Reinhard Farwig, An $ L^q$-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, $ L^q$-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 $ L^{q}$-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, $ L^q$ 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, $ L^q$-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, $ L^{q}$-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)

Similar Articles

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

Mads Kyed
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany

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

American Mathematical Society