On a mapping property of the Oseen operator with rotation

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October 2013, 6(5): 1315-1322. doi: 10.3934/dcdss.2013.6.1315

On a mapping property of the Oseen operator with rotation

1.	Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany

Received August 2011 Revised November 2011 Published March 2013

Abstract
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The Oseen problem arises as the linearization of a steady-state Navier-Stokes flow past a translating body. If the body, in addition to the translational motion, is also rotating, the corresponding linearization of the equations of motion, written in a frame attached to the body, yields the Oseen system with extra terms in the momentum equation due to the rotation. In this paper, the effect these rotation terms have on the asymptotic structure at spatial infinity of a solution to the system is studied. A mapping property of the whole space Oseen operator with rotation is identified from which asymptotic properties of a solution can be derived. As an application, an asymptotic expansion of a steady-state, linearized Navier-Stokes flow past a rotating and translating body is established.

Keywords: Oseen, Navier-Stokes, asymptotic structure., rotation.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 35B40, 35C20, 76D0.

Citation: Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315

References:

[1]	R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle,, Tohoku Math. J. (2), 58 (2006), 129.
[2]	R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle,, Manuscr. Math., 136 (2011), 315. doi: 10.1007/s00229-011-0479-0.
[3]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems,", Springer Tracts in Natural Philosophy, 38 (1994). doi: 10.1007/978-1-4612-5364-8.
[4]	G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653.
[5]	G. P. Galdi and M. Kyed, A simple proof of $L^q$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions,, Proc. Amer. Math. Soc., 141 (2013), 573. doi: 10.1090/S0002-9939-2012-11638-7.
[6]	G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable,, Arch. Ration. Mech. Anal., 200 (2011), 21. doi: 10.1007/s00205-010-0350-6.
[7]	M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, to appear in Q. Appl. Math., (2011).

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References:

[1]	R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle,, Tohoku Math. J. (2), 58 (2006), 129.
[2]	R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle,, Manuscr. Math., 136 (2011), 315. doi: 10.1007/s00229-011-0479-0.
[3]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems,", Springer Tracts in Natural Philosophy, 38 (1994). doi: 10.1007/978-1-4612-5364-8.
[4]	G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653.
[5]	G. P. Galdi and M. Kyed, A simple proof of $L^q$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions,, Proc. Amer. Math. Soc., 141 (2013), 573. doi: 10.1090/S0002-9939-2012-11638-7.
[6]	G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable,, Arch. Ration. Mech. Anal., 200 (2011), 21. doi: 10.1007/s00205-010-0350-6.
[7]	M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, to appear in Q. Appl. Math., (2011).

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