On a mapping property of the Oseen operator with rotation

Previous Article
Remarks on the theory of Oldroyd-B fluids in exterior domains
DCDS-S Home
This Issue
Next Article
A remark on the Stokes problem in Lorentz spaces

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
Related Papers

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, to appear in Q. Appl. Math., (2011). Google Scholar

show all references

References:

[1]	R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle,, Tohoku Math. J. (2), 58 (2006), 129. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications,, in, (2002), 653. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	M. Kyed, Asymptotic profile of a linearized flow past a rotating body,, to appear in Q. Appl. Math., (2011). Google Scholar

[1]	Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[2]	Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189
[3]	Trinh Viet Duoc. Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3387-3405. doi: 10.3934/dcds.2018145
[4]	Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
[5]	Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[6]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[7]	Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R³. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[8]	Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
[9]	Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245
[10]	Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[11]	Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[12]	Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[13]	G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[14]	Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
[15]	Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[16]	Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[17]	Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[18]	Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[19]	C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[20]	Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences