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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

 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 $L^{3,q}_{σ}$.

Citation: 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
##### References:
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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

##### 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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