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

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July 2018, 38(7): 3387-3405. doi: 10.3934/dcds.2018145

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

Received July 2017 Revised January 2018 Published April 2018

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

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: Trinh Viet Duoc. Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3387-3405. doi: 10.3934/dcds.2018145

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.

show all references

Dedicated to Professor Dang Dinh Chau on the occasion of his 70th birthday

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.

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