Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle

Trinh Viet Duoc

doi:10.3934/dcds.2018145

Article Contents

2018, Volume 38, Issue 7: 3387-3405. Doi: 10.3934/dcds.2018145

This issue Previous Article Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind Next Article On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type

Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle

Trinh Viet Duoc^,

Vietnam National University, Hanoi University of Science, Faculty of Mathematics, Mechanics, and Informatics, 334 Nguyen Trai, Hanoi, Vietnam

* Corresponding author: Trinh Viet Duoc
* Corresponding author: Trinh Viet Duoc

Received: July 2017

Revised: January 2018

Published: July 2018

This research is funded by the Vietnam National University, Hanoi (VNU) under project number QG.17.07.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we investigate Navier-Stokes-Oseen equation describing flows of incompressible viscous fluid passing a translating and rotating obstacle. The existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions to Navier-Stokes-Oseen equation in the solenoidal Lorentz space $ L^{3}_{σ, w} $ are shown. Moreover, we also prove the unique existence of time-local mild solutions to this equation in the solenoidal Lorentz spaces $ L^{3,q}_{σ} $.

Keywords:

Bounded and almost periodic weak mild solutions,
time-local mild solutions,
Navier-Stokes-Oseen equation,
Navier-Stokes equation,
Oseen operator,
rotating and translating obstacle.

Mathematics Subject Classification: Primary: 35B15, 35B35, 35Q30; Secondary: 76D07.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin-Heidelberg-New York, 1976. doi: 10.1007/978-3-642-66451-9.
[2]	W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382. doi: 10.1007/BF02392469.
[3]	W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in $ L^p $-spaces, Math. Z., 196 (1987), 415-425. doi: 10.1007/BF01200362.
[4]	R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer, 2016. doi: 10.1007/978-3-319-30034-4.
[5]	R. Farwig and T. Hishida, Stationary Navier-Stokes flows around a rotating obstacle, Funkc. Ekvac., 50 (2007), 371-403. doi: 10.1619/fesi.50.371.
[6]	G. P. Galdi, Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane, Discrete Continuous Dynam. Systems -S, 6 (2013), 1237-1257. doi: 10.3934/dcdss.2013.6.1237.
[7]	G. P. Galdi and A. L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific J. Math., 223 (2006), 251-267. doi: 10.2140/pjm.2006.223.251.
[8]	G. P. Galdi and A. L. Silvestre, On the motion of a rigid body in a Navier-Stokes liquid under the action of a time-periodic force, Indiana Univ. Math. J., 58 (2009), 2805-2842. doi: 10.1512/iumj.2009.58.3758.
[9]	G. P. Galdi and A. L. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rational Mech. Anal., 184 (2007), 371-400. doi: 10.1007/s00205-006-0026-4.
[10]	G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flows past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9.
[11]	M. Geissert, H. Heck and M. Hieber, $ L_p $-Theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.
[12]	Y. Giga, Solutions for semilinear parabolic equations in $ L^p $ and regurlarity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[13]	Y. Giga, S. Matsui and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3 (2001), 302-315. doi: 10.1007/PL00000973.
[14]	M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z., 265 (2010), 481-491. doi: 10.1007/s00209-009-0525-8.
[15]	M. Hieber and O. Sawada, The Navier-Stokes equations in $ \mathbb{R}^n $ with linearly growing initial data, Arch. Ration. Mech. Anal., 175 (2005), 269-285. doi: 10.1007/s00205-004-0347-0.
[16]	T. Hishida and Y. Shibata, $ L_p - L_q $ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.
[17]	T. Kato, Strong $ L^p $-solutions of Navier-Stokes equations in $ \mathbb{R}^n $ with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[18]	T. Kobayashi and Y. Shibata, On the Oseen equation in the three dimensional exterior domains, Math. Ann., 310 (1998), 1-45. doi: 10.1007/s002080050134.
[19]	H. Komatsu, A general interpolation theorem of Marcinkiewics type, Tôhoku Math. J., 33 (1981), 383-393. doi: 10.2748/tmj/1178229401.
[20]	M. Kyed, The existence and regularity of time-periodic solutions to the three dimensional Navier-Stokes equations in the whole plane, Nonlinearity, 27 (2014), 2909-2935. doi: 10.1088/0951-7715/27/12/2909.
[21]	B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge, 1982.
[22]	A. Lunardi, Interpolation Theory, Birkhäuser, 2009.
[23]	P. Maremonti, Existence and stability of time periodic solutions to the Navier-Stokes equations in exterior domains, J. Math. Sci., 93 (1999), 719-746. doi: 10.1007/BF02366850.
[24]	T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.
[25]	T. H. Nguyen, T. V. Duoc, T. N. H. Vu and T. M. Vu, Boundedness, almost periodicity and stability of certain Navier-Stokes flows in unbounded domains, J. Differential Equations, 263 (2017), 8979-9002. doi: 10.1016/j.jde.2017.08.061.
[26]	Y. Shibata, On a $ C^0 $ semigroup associated with a modified Oseen equation with rotating effect, Adv. Math. Fluid Mech, (2010), 513-551. doi: 10.1007/978-3-642-04068-9_29.
[27]	Y. Shibata, On the Oseen semigroup with rotating effect, Funct. Anal. Evol. Equ., (2008), 595-611. doi: 10.1007/978-3-7643-7794-6_36.
[28]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.
[29]	M. Yamazaki, The Navier-Stokes equations in the weak-$ L^n $ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.