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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 & 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.  Google Scholar

[2]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.  doi: 10.1007/BF02392469.  Google Scholar

[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.  Google Scholar

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R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer, 2016. doi: 10.1007/978-3-319-30034-4.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. Komatsu, A general interpolation theorem of Marcinkiewics type, Tôhoku Math. J., 33 (1981), 383-393.  doi: 10.2748/tmj/1178229401.  Google Scholar

[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.  Google Scholar

[21]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge, 1982.  Google Scholar

[22]

A. Lunardi, Interpolation Theory, Birkhäuser, 2009.  Google Scholar

[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.  Google Scholar

[24]

T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.  doi: 10.1007/BF02392469.  Google Scholar

[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.  Google Scholar

[4]

R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer, 2016. doi: 10.1007/978-3-319-30034-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. Komatsu, A general interpolation theorem of Marcinkiewics type, Tôhoku Math. J., 33 (1981), 383-393.  doi: 10.2748/tmj/1178229401.  Google Scholar

[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.  Google Scholar

[21]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge, 1982.  Google Scholar

[22]

A. Lunardi, Interpolation Theory, Birkhäuser, 2009.  Google Scholar

[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.  Google Scholar

[24]

T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.  Google Scholar

[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.  Google Scholar

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