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September  2019, 18(5): 2349-2376. doi: 10.3934/cpaa.2019106

Random attractor for the 2D stochastic nematic liquid crystals flows

1. 

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China

2. 

No. 174 Sha Zheng street, Shapingba District, Chongqing University, Chongqing 401331, China

* Corresponding author

Received  March 2018 Revised  December 2018 Published  April 2019

Fund Project: This work was partially supported by NNSF of China(Grant No. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universities(Grant No.2018CDXYST0024, 106112015CDJXY100005) and China Scholarship Council (Grant No.:201506055003)

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly non-linear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

Citation: Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042–5044. Google Scholar

[3]

Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Lukaszewicz and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919. doi: 10.1016/j.jde.2013.07.043. Google Scholar

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Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629. doi: 10.1090/S0002-9947-06-03923-7. Google Scholar

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Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Some results on a system of nonlinear SPDEs with multiplicative noise arising in the dynamics of nematic liquid crystals, arXiv: 1310.8641v3, [math.PR]2016.Google Scholar

[6]

Z. BrzeźniakE. Hausenblas and P. Razafimandimby, Stochastic nonparabolic dissipative systems modelling the flow of liquid crystals: strong solution, RIMS Kokyuroku Proceeding of RIMS Symposium on Mathematical Analysis of Incompressible Flow, 1875 (2014), 41-72. Google Scholar

[7]

Z. Brzeźniak, U. Manna and P. Akash Ashirbad, Existence of weak martingale solution of nematic liquid crystals driven by pure jump noise in the Marcus canonical form, J. Differential Equations, (2018). doi: 10.1016/j.jde.2018.11.001. Google Scholar

[8]

H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 37 (1991), 153-173. doi: 10.1080/17442509108833733. Google Scholar

[9]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar

[11]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications vol 44, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[12]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. Google Scholar

[13]

N. Gershenfeld, The Nature of Mathematical Modeling, Cambridge Press, Cambridge, 1999. Google Scholar

[14]

B. Guo and D. Huang, 3d stochastic primitive equations of the large-scale ocean: global wellposedness and attractors, Commun. Math. Phys., 286 (2009), 697-723. doi: 10.1007/s00220-008-0654-7. Google Scholar

[15]

M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992. doi: 10.1007/s00033-011-0157-9. Google Scholar

[16]

W. HorsthemkeC. R. DoeringR. Lefever and A. S. Chi, Effect of external-field fluctuations on instabilities in nematic liquid crystals, Phys. Rev. A, 31 (1985), 1123-1135. Google Scholar

[17]

W. Horsthemke and R. Lefever, Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984. Google Scholar

[18]

R. M. JendrejackJ. J. de Pablo and M. D. Graham, A method for multiscale simulation of flowing complex fluids, J. Non-Newtonian Fluid Mech., 108 (2002), 123-142. Google Scholar

[19]

Y. LiZ. Brzeźniak and J. Z. Zhou, Conceptual analysis and random attractor for dissipative random dynamical systems, Acta Math. Sci. Ser. B (Engl. Ed.), 28 (2008), 253-268. doi: 10.1016/S0252-9602(08)60026-0. Google Scholar

[20]

F. M. Leslie, Theory of flow phenomena in liquid crystals, in, Advances in Liquid Crystals (eds. G. Brown), Academic Press, 4 (1979), 1–81.Google Scholar

[21]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transitions and flow phenomena, Comm. Pure Appl. Math., XLII (1989), 789-814. doi: 10.1002/cpa.3160420605. Google Scholar

[22]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of Liquid Crystals, Communications on Pure and Applied Mathematics, XLVIII (1995), 501-537. doi: 10.1002/cpa.3160480503. Google Scholar

[23]

F.-H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-23. doi: 10.3934/dcds.2011.31.1. Google Scholar

[24]

J. Lions, Quelques Méthode de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar

[25]

K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004.Google Scholar

[26]

M. Laso and H. C. Öttinger, Calculation of viscoelastic flow using molecular models: the connffessit approach, J. Non-Newtonian Fluid Mech., 1 (1993), 1-20. Google Scholar

[27]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142997327282. Google Scholar

[28]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, M2AN Math. Model. Numer. Anal., 36 (2002), 205-222. doi: 10.1051/m2an:2002010. Google Scholar

[29]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. Google Scholar

[30]

S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Communications in Partial Differential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. Google Scholar

[31]

M. Somasi and B. Khomami, Linear stability and dynamics of viscoelastic flows using time-dependent stochastic simulation techniques, Journal of Non-Newtonian Fluid Mechanics, 93 (2000), 339-362. Google Scholar

[32]

M. Somasi and B. Khomami, A new approach for studying the hydrodynamic stability of fluids with microstructure, Phys Fluids, 13 (2001), 1811-1814. Google Scholar

[33]

M. San Miguel, Nematic liquid crystals in a stochastic magnetic field: Spatial correlations, Phys. Rev. A, 32 (1985), 3811–3813, 1985.Google Scholar

[34]

F. Sagués and M. San Miguel, Dynamics of Fréedericksz transition in a fluctuating magnetic field, Phys. Rev. A., 32 (1985), 1843-1851. Google Scholar

[35]

Guoli Zhou, Random attractor of the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637. doi: 10.1016/j.jde.2018.12.009. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

[2]

J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042–5044. Google Scholar

[3]

Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Lukaszewicz and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919. doi: 10.1016/j.jde.2013.07.043. Google Scholar

[4]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629. doi: 10.1090/S0002-9947-06-03923-7. Google Scholar

[5]

Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Some results on a system of nonlinear SPDEs with multiplicative noise arising in the dynamics of nematic liquid crystals, arXiv: 1310.8641v3, [math.PR]2016.Google Scholar

[6]

Z. BrzeźniakE. Hausenblas and P. Razafimandimby, Stochastic nonparabolic dissipative systems modelling the flow of liquid crystals: strong solution, RIMS Kokyuroku Proceeding of RIMS Symposium on Mathematical Analysis of Incompressible Flow, 1875 (2014), 41-72. Google Scholar

[7]

Z. Brzeźniak, U. Manna and P. Akash Ashirbad, Existence of weak martingale solution of nematic liquid crystals driven by pure jump noise in the Marcus canonical form, J. Differential Equations, (2018). doi: 10.1016/j.jde.2018.11.001. Google Scholar

[8]

H. Crauel, Markov measures for random dynamical systems, Stochastics Stochastics Rep., 37 (1991), 153-173. doi: 10.1080/17442509108833733. Google Scholar

[9]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar

[11]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications vol 44, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[12]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. Google Scholar

[13]

N. Gershenfeld, The Nature of Mathematical Modeling, Cambridge Press, Cambridge, 1999. Google Scholar

[14]

B. Guo and D. Huang, 3d stochastic primitive equations of the large-scale ocean: global wellposedness and attractors, Commun. Math. Phys., 286 (2009), 697-723. doi: 10.1007/s00220-008-0654-7. Google Scholar

[15]

M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow, Z. Angew. Math. Phys., 62 (2011), 979-992. doi: 10.1007/s00033-011-0157-9. Google Scholar

[16]

W. HorsthemkeC. R. DoeringR. Lefever and A. S. Chi, Effect of external-field fluctuations on instabilities in nematic liquid crystals, Phys. Rev. A, 31 (1985), 1123-1135. Google Scholar

[17]

W. Horsthemke and R. Lefever, Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984. Google Scholar

[18]

R. M. JendrejackJ. J. de Pablo and M. D. Graham, A method for multiscale simulation of flowing complex fluids, J. Non-Newtonian Fluid Mech., 108 (2002), 123-142. Google Scholar

[19]

Y. LiZ. Brzeźniak and J. Z. Zhou, Conceptual analysis and random attractor for dissipative random dynamical systems, Acta Math. Sci. Ser. B (Engl. Ed.), 28 (2008), 253-268. doi: 10.1016/S0252-9602(08)60026-0. Google Scholar

[20]

F. M. Leslie, Theory of flow phenomena in liquid crystals, in, Advances in Liquid Crystals (eds. G. Brown), Academic Press, 4 (1979), 1–81.Google Scholar

[21]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transitions and flow phenomena, Comm. Pure Appl. Math., XLII (1989), 789-814. doi: 10.1002/cpa.3160420605. Google Scholar

[22]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of Liquid Crystals, Communications on Pure and Applied Mathematics, XLVIII (1995), 501-537. doi: 10.1002/cpa.3160480503. Google Scholar

[23]

F.-H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-23. doi: 10.3934/dcds.2011.31.1. Google Scholar

[24]

J. Lions, Quelques Méthode de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. Google Scholar

[25]

K. Liu, Stability of Stochastic Differential Equations in Infinite Dimensions, Springer Verlag, New York, 2004.Google Scholar

[26]

M. Laso and H. C. Öttinger, Calculation of viscoelastic flow using molecular models: the connffessit approach, J. Non-Newtonian Fluid Mech., 1 (1993), 1-20. Google Scholar

[27]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142997327282. Google Scholar

[28]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, M2AN Math. Model. Numer. Anal., 36 (2002), 205-222. doi: 10.1051/m2an:2002010. Google Scholar

[29]

J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. Google Scholar

[30]

S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Communications in Partial Differential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. Google Scholar

[31]

M. Somasi and B. Khomami, Linear stability and dynamics of viscoelastic flows using time-dependent stochastic simulation techniques, Journal of Non-Newtonian Fluid Mechanics, 93 (2000), 339-362. Google Scholar

[32]

M. Somasi and B. Khomami, A new approach for studying the hydrodynamic stability of fluids with microstructure, Phys Fluids, 13 (2001), 1811-1814. Google Scholar

[33]

M. San Miguel, Nematic liquid crystals in a stochastic magnetic field: Spatial correlations, Phys. Rev. A, 32 (1985), 3811–3813, 1985.Google Scholar

[34]

F. Sagués and M. San Miguel, Dynamics of Fréedericksz transition in a fluctuating magnetic field, Phys. Rev. A., 32 (1985), 1843-1851. Google Scholar

[35]

Guoli Zhou, Random attractor of the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637. doi: 10.1016/j.jde.2018.12.009. Google Scholar

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