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July  2020, 19(7): 3805-3827. doi: 10.3934/cpaa.2020168

Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations

Xiuqing Wang 1, , Yuming Qin 2,, and  Alain Miranville 3,,

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

*Corresponding author

Received  November 2019 Revised  February 2020 Published  April 2020

Fund Project: This paper was supported in part by the NNSF of China with contract numbers 11671075, 11801068, 11971110 and the Graduate Innovation Fund Project of Donghua University with contract number CUSF-DH-D-2020077

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^{+} $.

Keywords: Liquid crystal flow, trajectory attractors, weak solutions.
Mathematics Subject Classification: Primary: 76N10, 76N15; Secondary: 35M13, 35Q35, 53C35.
Citation: Xiuqing Wang, Yuming Qin, Alain Miranville. Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3805-3827. doi: 10.3934/cpaa.2020168
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Collouium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Vol. 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[10]

W. E, Nonlinear continuum theory of smectic-A liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175.  doi: 10.1007/s002050050026.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[13]

P. G. de Gennes, Viscous flows in smectic A liquid crystals, Phys. Fluids, 17 (1974), 1645. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. M. Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017. doi: 10.1007/978-3-319-00831-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Collouium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Vol. 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[10]

W. E, Nonlinear continuum theory of smectic-A liquid crystals, Arch. Ration. Mech. Anal., 137 (1997), 159-175.  doi: 10.1007/s002050050026.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. G. de Gennes, Viscous flows in smectic A liquid crystals, Phys. Fluids, 17 (1974), 1645. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. M. Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017. doi: 10.1007/978-3-319-00831-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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