# American Institute of Mathematical Sciences

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

## On a mapping property of the Oseen operator with rotation

Received  August 2011 Revised  November 2011 Published  March 2013

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.
Citation: Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete and 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), 129-147. [2] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscr. Math., 136 (2011), 315-338. 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, Springer-Verlag, New York, 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 "Handbook of Mathematical Fluid Dynamics. Vol. 1" (eds. S. Friedlander, et al.), North-Holland, Amsterdam, (2002), 653-791. [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), 573-583. 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), 21-58. 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), 129-147. [2] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscr. Math., 136 (2011), 315-338. 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, Springer-Verlag, New York, 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 "Handbook of Mathematical Fluid Dynamics. Vol. 1" (eds. S. Friedlander, et al.), North-Holland, Amsterdam, (2002), 653-791. [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), 573-583. 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), 21-58. 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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