A simple proof of estimates for the steadystate 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), 13131322
MSC (2010):
Primary 35Q30, 35B45, 76D07
Published electronically:
August 23, 2012
MathSciNet review:
3008878
Fulltext 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 steadystate threedimensional 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 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,
Pacific J. Math. 215 (2004), no. 2, 297–312. MR 2068783
(2005f:35078), 10.2140/pjm.2004.215.297
 [Gal94]
Giovanni
P. Galdi, An introduction to the mathematical theory of the
NavierStokes equations. Vol. I, Springer Tracts in Natural
Philosophy, vol. 38, SpringerVerlag, New York, 1994. Linearized
steady problems. MR 1284205
(95i:35216a)
 [Gal02]
Giovanni
P. Galdi, On the motion of a rigid body in a viscous liquid: a
mathematical analysis with applications, Handbook of mathematical
fluid dynamics, Vol. I, NorthHolland, Amsterdam, 2002,
pp. 653–791. MR 1942470
(2003j:76024)
 [Gal03]
Giovanni
P. Galdi, Steady flow of a NavierStokes fluid around a rotating
obstacle, J. Elasticity 71 (2003), no. 13,
1–31. Essays and papers dedicated to the memory of Clifford Ambrose
Truesdell III, Vol. II. MR 2042672
(2005c:76030), 10.1023/B:ELAS.0000005543.00407.5e
 [GK11a]
Giovanni P. Galdi and Mads Kyed, A simple proof of estimates for the steadystate Oseen and Stokes equations in a rotating frame. Part I: Strong solutions, 2011. To appear in Proc. Amer. Math. Soc.
 [GK11b]
Giovanni
P. Galdi and Mads
Kyed, Steadystate NavierStokes flows past a rotating body: Leray
solutions are physically reasonable, Arch. Ration. Mech. Anal.
200 (2011), no. 1, 21–58. MR 2781585
(2012c:35324), 10.1007/s0020501003506
 [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]
Joanna
Rencławowicz and Wojciech
M. Zajączkowski (eds.), Parabolic and NavierStokes
equations. Part 1, Banach Center Publications, vol. 81, Polish
Academy of Sciences, Institute of Mathematics, Warsaw, 2008. Papers from
the conference held in Będlewo, September 10–17, 2006. MR 2547463
(2010d:35002)
 [KNP10]
S.
Kračmar and Š.
Nečasová, 𝐿^{𝑞}approach of weak
solutions to stationary rotating Oseen equations in exterior
domains, Quart. Appl. Math.
68 (2010), no. 3,
421–437. MR 2676969
(2012b:76032), 10.1090/S0033569X10012104
 [Lad69]
O.
A. Ladyzhenskaya, The mathematical theory of viscous incompressible
flow, Second English edition, revised and enlarged. Translated from
the Russian by Richard A. Silverman and John Chu. Mathematics and its
Applications, Vol. 2, Gordon and Breach, Science Publishers, New
YorkLondonParis, 1969. MR 0254401
(40 #7610)
 [Sil04]
Ana
Leonor Silvestre, On the existence of steady flows of a
NavierStokes liquid around a moving rigid body, Math. Methods Appl.
Sci. 27 (2004), no. 12, 1399–1409. MR 2069156
(2005f:35251), 10.1002/mma.509
 [Far06]
 Reinhard Farwig, An analysis of viscous fluid flow past a rotating obstacle, Tohoku Math. J. (2) 58 (2006), no. 1, 129147. 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, 297312. MR 2068783 (2005f:35078)
 [Gal94]
 Giovanni P. Galdi, An introduction to the mathematical theory of the NavierStokes equations. Vol. I: Linearized steady problems, Springer Tracts in Natural Philosophy. 38. New York: SpringerVerlag, 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. 653791, 2002. MR 1942470 (2003j:76024)
 [Gal03]
 , Steady flow of a NavierStokes fluid around a rotating obstacle, Journal of Elasticity 71 (2003), 131. MR 2042672 (2005c:76030)
 [GK11a]
 Giovanni P. Galdi and Mads Kyed, A simple proof of estimates for the steadystate Oseen and Stokes equations in a rotating frame. Part I: Strong solutions, 2011. To appear in Proc. Amer. Math. Soc.
 [GK11b]
 , Steadystate NavierStokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 2158. 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, 743767. 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 NavierStokes equations. Part 1. Proceedings of the conference, Bedlewo, Poland, September 1017, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 259276, 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, 421437. 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 NavierStokes liquid around a moving rigid body, Math. Methods Appl. Sci. 27 (2004), no. 12, 13991409. 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
Email:
galdi@pitt.edu
Mads Kyed
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D64289 Darmstadt, Germany
Email:
kyed@mathematik.tudarmstadt.de
DOI:
http://dx.doi.org/10.1090/S000299392012116405
Received by editor(s):
August 9, 2011
Published electronically:
August 23, 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
