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Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities
Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations
1. | College of Information Science and Technology, Donghua University, Shanghai 201620, China |
2. | Department of Mathematics, Institute for Nonlinear Sciences, Donghua University, Shanghai 201620, China |
3. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7648, SP2MI, Boulevard Marie et Pierre Curie-Téléport 2, F–6962 Chasseneuil Futuroscope Cedex, France |
In this paper, we first establish the existence of trajectory attractors for the 3D smectic-A liquid crystal flow system and 3D smectic-A liquid crystal flow-$ \alpha $ model, and then prove that the latter trajectory attractor converges to the former one as the parameter $ \alpha\rightarrow 0^{+} $.
References:
[1] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On Convergence of Trajectory Attractors of 3D Navier-Stokes-$\alpha$ Model as $\alpha$ Approaches 0, Russian Academy of Sciences, (DoM) and London Mathematical Society, 2007.
doi: 10.1070/SM2007v198n12ABEH003902. |
[2] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik,
On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52.
doi: 10.3934/dcds.2007.17.481. |
[3] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory attractors for evolution Equations, C. R. Acad. Sci. Paris. Ser. I, 321 (1995), 1309-1314.
doi: 10.1016/S0021-7824(97)89978-3. |
[4] |
V. V. Chepyzhov and M.I. Vishik,
Evolution Equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.
doi: 10.1016/S0021-7824(97)89978-3. |
[5] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Collouium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002. |
[7] |
B. Climent-Ezquerra and F. Guill$\acute{e}$n-Gonz$\acute{a}$lez,
Global in time solution and time-periodicity for a Smectic-A liquid crystal model, Commun. Pure Appl. Anal., 9 (2010), 1473-1493.
doi: 10.3934/cpaa.2010.9.1473. |
[8] |
G. Deugoue,
Approximation of the trajectory attractor of the 3D MHD system, Commun. Pure Appl. Anal., 12 (2013), 2119-2144.
doi: 10.3934/cpaa.2013.12.2119. |
[9] |
L. C. Evans, Partial Differential Equations, Vol. 49, American Mathematical Society, Providence, RI, 1997. |
[10] |
W. E,
Nonlinear continuum theory of smectic-A liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175.
doi: 10.1007/s002050050026. |
[11] |
S. Frigeri and E. Rocca,
Trajectory attractors for the Sun-Liu model for nematic liquid crystals in 3D, Nonlinearity, 26 (2013), 933-958.
doi: 10.1088/0951-7715/26/4/933. |
[12] |
C. G. Gal and M. Grasselli,
Trajectory Attractors for Binary Fluid Mixtures in 3D, Chin. Ann. Math., 31B (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[13] |
P. G. de Gennes, Viscous flows in smectic A liquid crystals, Phys. Fluids, 17 (1974), 1645. |
[14] |
M. Grasselli, G. Schimperna and S. Zelik,
Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-738.
doi: 10.1088/0951-7715/23/3/016. |
[15] |
A. B. Liu and C. C. Liu,
Global attractor for a smectic-A liquid crystal model in 2D, Boll. Unione Mat. Ital., 11 (2018), 581-594.
doi: 10.1007/s40574-018-0156-2. |
[16] |
C. Liu,
The dynamic for incompressible Smectic-A liquid crystals: existence and regularity, Discrete Contin. Dyn. Syst., 6 (2000), 591-608.
doi: 10.3934/dcds.2000.6.591. |
[17] |
F. H. Lin and C. Liu,
Non-parabolic dissipative systems modelling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[18] |
Y. M. Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017.
doi: 10.1007/978-3-319-00831-8. |
[19] |
A. Segatti and H. Wu,
Finite dimensional reduction and convergence to equilibrium for incompressible smectic-A liquid crystal flows, SIAM J. Math. Anal., 43 (2011), 2445-2481.
doi: 10.1137/100813427. |
[20] |
I. W. Stewart,
Dynamic theory for smectic A liquid crystals, Continuum Mech. Thermodyn., 18 (2007), 343-360.
doi: 10.1007/s00161-006-0035-4. |
[21] |
R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[22] |
X. Q. Wang and Y. M. Qin, Upper semicontinuity of trajectory attractors for 3D incompressible Navier-Stokes equation, Appl. Math. Optim., (2019).
doi: 10.1007/s00245-019-09625-7. |
[23] |
M. C. Zelati and C. G. Gal, Singular limits of Voight models in fluid dynamics, J. Math Fluid Mech., 17 (2005), 233-259.
doi: 10.1007/s00021-015-0201-1. |
show all references
References:
[1] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On Convergence of Trajectory Attractors of 3D Navier-Stokes-$\alpha$ Model as $\alpha$ Approaches 0, Russian Academy of Sciences, (DoM) and London Mathematical Society, 2007.
doi: 10.1070/SM2007v198n12ABEH003902. |
[2] |
V. V. Chepyzhov, E. S. Titi and M. I. Vishik,
On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 33-52.
doi: 10.3934/dcds.2007.17.481. |
[3] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory attractors for evolution Equations, C. R. Acad. Sci. Paris. Ser. I, 321 (1995), 1309-1314.
doi: 10.1016/S0021-7824(97)89978-3. |
[4] |
V. V. Chepyzhov and M.I. Vishik,
Evolution Equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.
doi: 10.1016/S0021-7824(97)89978-3. |
[5] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory and global attractors of three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193.
doi: 10.1023/A:1014190629738. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Collouium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002. |
[7] |
B. Climent-Ezquerra and F. Guill$\acute{e}$n-Gonz$\acute{a}$lez,
Global in time solution and time-periodicity for a Smectic-A liquid crystal model, Commun. Pure Appl. Anal., 9 (2010), 1473-1493.
doi: 10.3934/cpaa.2010.9.1473. |
[8] |
G. Deugoue,
Approximation of the trajectory attractor of the 3D MHD system, Commun. Pure Appl. Anal., 12 (2013), 2119-2144.
doi: 10.3934/cpaa.2013.12.2119. |
[9] |
L. C. Evans, Partial Differential Equations, Vol. 49, American Mathematical Society, Providence, RI, 1997. |
[10] |
W. E,
Nonlinear continuum theory of smectic-A liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175.
doi: 10.1007/s002050050026. |
[11] |
S. Frigeri and E. Rocca,
Trajectory attractors for the Sun-Liu model for nematic liquid crystals in 3D, Nonlinearity, 26 (2013), 933-958.
doi: 10.1088/0951-7715/26/4/933. |
[12] |
C. G. Gal and M. Grasselli,
Trajectory Attractors for Binary Fluid Mixtures in 3D, Chin. Ann. Math., 31B (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[13] |
P. G. de Gennes, Viscous flows in smectic A liquid crystals, Phys. Fluids, 17 (1974), 1645. |
[14] |
M. Grasselli, G. Schimperna and S. Zelik,
Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-738.
doi: 10.1088/0951-7715/23/3/016. |
[15] |
A. B. Liu and C. C. Liu,
Global attractor for a smectic-A liquid crystal model in 2D, Boll. Unione Mat. Ital., 11 (2018), 581-594.
doi: 10.1007/s40574-018-0156-2. |
[16] |
C. Liu,
The dynamic for incompressible Smectic-A liquid crystals: existence and regularity, Discrete Contin. Dyn. Syst., 6 (2000), 591-608.
doi: 10.3934/dcds.2000.6.591. |
[17] |
F. H. Lin and C. Liu,
Non-parabolic dissipative systems modelling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[18] |
Y. M. Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017.
doi: 10.1007/978-3-319-00831-8. |
[19] |
A. Segatti and H. Wu,
Finite dimensional reduction and convergence to equilibrium for incompressible smectic-A liquid crystal flows, SIAM J. Math. Anal., 43 (2011), 2445-2481.
doi: 10.1137/100813427. |
[20] |
I. W. Stewart,
Dynamic theory for smectic A liquid crystals, Continuum Mech. Thermodyn., 18 (2007), 343-360.
doi: 10.1007/s00161-006-0035-4. |
[21] |
R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[22] |
X. Q. Wang and Y. M. Qin, Upper semicontinuity of trajectory attractors for 3D incompressible Navier-Stokes equation, Appl. Math. Optim., (2019).
doi: 10.1007/s00245-019-09625-7. |
[23] |
M. C. Zelati and C. G. Gal, Singular limits of Voight models in fluid dynamics, J. Math Fluid Mech., 17 (2005), 233-259.
doi: 10.1007/s00021-015-0201-1. |
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