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Robust control of a Cahn-Hilliard-Navier-Stokes model
Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics
1. | School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi |
2. | School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vietnam |
3. | Faculty of Information Technology, Department of Mathematics, Water Resources University, Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Hanoi, Vietnam |
References:
[1] |
J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin-Heidelberg-NewYork, 1976. |
[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] |
R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
doi: 10.1007/BF02571437. |
[4] |
R. Farwig and T. Hishida, Stationary Navier-Stokes flows around a rotating obstacle, Funkcial. Ekvac., 50 (2007), 371-403.
doi: 10.1619/fesi.50.371. |
[5] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38, Springer, Berlin, Heidelberg, New York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[6] |
G. P. Galdi, J. G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest, Arch. Ration. Mech. Anal., 138 (1997), 307-318.
doi: 10.1007/s002050050043. |
[7] |
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. |
[8] |
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. |
[9] |
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[10] |
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. Rational Mech. Anal., 193 (2009), 339-421.
doi: 10.1007/s00205-008-0130-8. |
[11] |
Nguyen Thieu Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, Journal of Differential Equations, 246 (2009), 1820-1844.
doi: 10.1016/j.jde.2008.10.010. |
[12] |
Nguyen Thieu Huy, Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle, Arch. Ration. Mech. Anal., 213 (2014), 689-703.
doi: 10.1007/s00205-014-0744-y. |
[13] |
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. |
[14] |
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. |
[15] |
H. Komatsu, A general interpolation theorem of Marcinkiewicz type, Tôhoku Math. J., 33 (1981), 383-393.
doi: 10.2748/tmj/1178229401. |
[16] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969. |
[17] |
P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. |
[18] |
P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space, Ric. Mat., 40 (1991), 81-135. |
[19] |
P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci., (New York), 93 (1999), 719-746.
doi: 10.1007/BF02366850. |
[20] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[21] |
F. K. G. Odqvist, Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten, Math. Z., 32 (1930), 329-375.
doi: 10.1007/BF01194638. |
[22] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978. |
[23] |
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-NewYork, 1976. |
[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] |
R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
doi: 10.1007/BF02571437. |
[4] |
R. Farwig and T. Hishida, Stationary Navier-Stokes flows around a rotating obstacle, Funkcial. Ekvac., 50 (2007), 371-403.
doi: 10.1619/fesi.50.371. |
[5] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38, Springer, Berlin, Heidelberg, New York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[6] |
G. P. Galdi, J. G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest, Arch. Ration. Mech. Anal., 138 (1997), 307-318.
doi: 10.1007/s002050050043. |
[7] |
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. |
[8] |
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. |
[9] |
Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[10] |
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. Rational Mech. Anal., 193 (2009), 339-421.
doi: 10.1007/s00205-008-0130-8. |
[11] |
Nguyen Thieu Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, Journal of Differential Equations, 246 (2009), 1820-1844.
doi: 10.1016/j.jde.2008.10.010. |
[12] |
Nguyen Thieu Huy, Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle, Arch. Ration. Mech. Anal., 213 (2014), 689-703.
doi: 10.1007/s00205-014-0744-y. |
[13] |
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. |
[14] |
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. |
[15] |
H. Komatsu, A general interpolation theorem of Marcinkiewicz type, Tôhoku Math. J., 33 (1981), 383-393.
doi: 10.2748/tmj/1178229401. |
[16] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969. |
[17] |
P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. |
[18] |
P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space, Ric. Mat., 40 (1991), 81-135. |
[19] |
P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci., (New York), 93 (1999), 719-746.
doi: 10.1007/BF02366850. |
[20] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[21] |
F. K. G. Odqvist, Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten, Math. Z., 32 (1930), 329-375.
doi: 10.1007/BF01194638. |
[22] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978. |
[23] |
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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