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Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind
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 |
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} $.
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
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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