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September  2019, 18(5): 2243-2264. doi: 10.3934/cpaa.2019101

On the existence and uniqueness of solution to a stochastic simplified liquid crystal model

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA

Received  August 2017 Revised  August 2017 Published  April 2019

We study in this article a stochastic version of a 2D simplified Ericksen-Leslie systems, which model the dynamic of nematic liquid crystals under the influence of stochastic external forces. We prove the existence and uniqueness of strong solution. The proof relies on a new formulation of the model proposed in [19] as well as a Galerkin approximation

Citation: T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101
References:
[1]

A. Bensoussan and R. Temam, Equations stochastiques de type Navier-Stokes, Journal of Functional Analysis, 13 (1973), 195-222. doi: 10.1016/0022-1236(73)90045-1.

[2]

H. Breckner, Galerkin approximation and the strong solution of the navier-stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239-259. doi: 10.1155/S1048953300000228.

[3]

Z. BrzeźiakW Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310. doi: 10.1016/j.nonrwa.2013.12.005.

[4]

Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, arXiv: 1310.8641, 2016.

[5]

Z. BrzeźniakE. Hausenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139. doi: 10.1016/j.na.2012.10.011.

[6]

Z. Brzeźniak, U. Manna and A. A. Panda, Existence of weak martingale solution of nematic liquid crystals driven by pure jump noise, 2017.

[7]

T. CaraballoJ. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2066), 459-479. doi: 10.1098/rspa.2005.1574.

[8]

C. CavaterraE. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57. doi: 10.1016/j.jde.2013.03.009.

[9]

Z. Dong and Y. Xie, Global solutions of stochastic 2D navier-stokes equations with lévy noise, Science in China Series A: Mathematics, 52 (2009), 1497-1524. doi: 10.1007/s11425-009-0124-5.

[10]

Z. Dong and J. Zhai, Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump, Journal of Differential Equations, 250 (2011), 2737-2778. doi: 10.1016/j.jde.2011.01.018.

[11]

P. A. Razafimandimby E. Hausenblas and M. Sango, Martingale solution to equations for differential type fluids of grade two driven by random force of lévy type, Potential Analysis, 38 (2013), 1291-1331. doi: 10.1007/s11118-012-9316-7.

[12]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.

[13]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358.

[14]

J. Fan and F. Jiang, Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions, Commun. Pure Appl. Anal., 15 (2016), 73-90. doi: 10.3934/cpaa.2016.15.73.

[15]

W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A, 15 (1982), 3025-3055.

[16]

E. FeireislM. Frémond and E. Rocca, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4.

[17]

E. FeireislE. Rocca and G. Schimperna, On a non-isothermal model for nematic liquid crystals, Nonlinearity, 24 (2011), 243-257. doi: 10.1088/0951-7715/24/1/012.

[18]

C. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[19]

H. GongJ. HuangL. Liu and X. Liu, Global strong solutions of the 2D simplified Ericksen-Leslie system, Nonlinearity, 28 (2015), 3677-3694. doi: 10.1088/0951-7715/28/10/3677.

[20]

H. GongJ. Li and C. Xu, Local well-posedness of strong solutions to density-dependent liquid crystal system, Nonlinear Anal., 147 (2016), 26-44. doi: 10.1016/j.na.2016.08.014.

[21]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[22]

M. C. HongJ. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in R3, Comm. Partial Differential Equations, 39 (2014), 1284-1328. doi: 10.1080/03605302.2013.871026.

[23]

M. C. Hong and Z. P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in R2, Adv. Math., 231 (2012), 1364-1400. doi: 10.1016/j.aim.2012.06.009.

[24]

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.

[25]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in R2, Comm. Math. Phys., 331 (2014), 805-850. doi: 10.1007/s00220-014-2079-9.

[26]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[27]

F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357.

[28]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[29]

J. LiE. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in R2, Math. Models Methods Appl. Sci., 26 (2016), 803-822. doi: 10.1142/S0218202516500184.

[30]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938. doi: 10.1007/s11401-010-0612-5.

[31]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583.

[32]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[33]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[34]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. doi: 10.3934/dcds.2011.31.1.

[35]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102.

[36]

F. H. LinJ. Y. Liu and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[37]

J. L. Lions and G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.

[38]

H. Breckner (Lisei), Approximation and optimal control of the stochastic Navier-Stokes equations, Dissertation, Martin-Luther University, Halle-Wittenberg, 1999. doi: 10.1080/02331930108844518.

[39]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012.

[40]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014.

[41]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 262 (2017), 1028-1054. doi: 10.1016/j.jde.2017.03.008.

[42]

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

[43]

E. Motyl, Martingale solution to the 2D and 3D stochastic Navier-Stokes equations driven by the compensated poisson random measure, Department of Mathematics and Computer Sciences, Lodz University, Preprint 13, 2011. doi: 10.1007/s11118-012-9300-2.

[44]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Analysis, 38 (2013), 863-912. doi: 10.1007/s11118-012-9300-2.

[45]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains- Abstract framework and applications, Stochastic Processes and their Applications, 124 (2014), 2052-2097. doi: 10.1016/j.spa.2014.01.009.

[46]

E. Pardoux, Equations and Dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse Université Paris XI, 1975.

[47]

S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, An evolution equation approach, in Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[48]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 152, Cambridge University Press, Cambridge, 2 edition, 2014. doi: 10.1017/CBO9781107295513.

[49]

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

[50]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455.

[51]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[52]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. doi: 10.1090/chel/343.

[53]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, In S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, Vol. Ⅲ, pages 535–658. Elsevier, 2004.

[54]

M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962. doi: 10.1007/s00526-013-0700-y.

[55]

M. WangW. Wang and Z. Zhang, On the uniqueness of weak solution for the 2-D Ericksen-Leslie system, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 919-941. doi: 10.3934/dcdsb.2016.21.919.

[56]

H. WuX. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.

[57]

H. WuX. Xu and C. Liu, On the general ericksen-leslie system: Parodi's relation, well-posedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107. doi: 10.1007/s00205-012-0588-2.

[58]

T. Xu and T. Zhang, Large deviation principles for 2D stochastic Navier-Stokes equations driven by lévy processes, Journal of Functional Analysis, 257 (2009), 1519-1545. doi: 10.1016/j.jfa.2009.05.007.

[59]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028.

[60]

W. V. LiZ. Dong and J. Zhai, Stationary weak solutions for stochastic 3d navier-stokes equations with lévy noise, Stochastic and Dynamics, 12 (2012), 1150006. doi: 10.1142/S0219493712003559.

[61]

J. Zhai and T. Zhang, Large deviations for 2D stochastic Navier-Stokes equations driven by multiplicative lévy noises, Bernoulli, 21 (2015), 2351-239. doi: 10.3150/14-BEJ647.

show all references

References:
[1]

A. Bensoussan and R. Temam, Equations stochastiques de type Navier-Stokes, Journal of Functional Analysis, 13 (1973), 195-222. doi: 10.1016/0022-1236(73)90045-1.

[2]

H. Breckner, Galerkin approximation and the strong solution of the navier-stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239-259. doi: 10.1155/S1048953300000228.

[3]

Z. BrzeźiakW Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310. doi: 10.1016/j.nonrwa.2013.12.005.

[4]

Z. Brzeźniak, E. Hausenblas and P. A. Razafimandimby, Some results on the penalised nematic liquid crystals driven by multiplicative noise, arXiv: 1310.8641, 2016.

[5]

Z. BrzeźniakE. Hausenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139. doi: 10.1016/j.na.2012.10.011.

[6]

Z. Brzeźniak, U. Manna and A. A. Panda, Existence of weak martingale solution of nematic liquid crystals driven by pure jump noise, 2017.

[7]

T. CaraballoJ. Real and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2066), 459-479. doi: 10.1098/rspa.2005.1574.

[8]

C. CavaterraE. Rocca and H. Wu, Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows, J. Differential Equations, 255 (2013), 24-57. doi: 10.1016/j.jde.2013.03.009.

[9]

Z. Dong and Y. Xie, Global solutions of stochastic 2D navier-stokes equations with lévy noise, Science in China Series A: Mathematics, 52 (2009), 1497-1524. doi: 10.1007/s11425-009-0124-5.

[10]

Z. Dong and J. Zhai, Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump, Journal of Differential Equations, 250 (2011), 2737-2778. doi: 10.1016/j.jde.2011.01.018.

[11]

P. A. Razafimandimby E. Hausenblas and M. Sango, Martingale solution to equations for differential type fluids of grade two driven by random force of lévy type, Potential Analysis, 38 (2013), 1291-1331. doi: 10.1007/s11118-012-9316-7.

[12]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34. doi: 10.1122/1.548883.

[13]

J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358.

[14]

J. Fan and F. Jiang, Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions, Commun. Pure Appl. Anal., 15 (2016), 73-90. doi: 10.3934/cpaa.2016.15.73.

[15]

W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A, 15 (1982), 3025-3055.

[16]

E. FeireislM. Frémond and E. Rocca, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4.

[17]

E. FeireislE. Rocca and G. Schimperna, On a non-isothermal model for nematic liquid crystals, Nonlinearity, 24 (2011), 243-257. doi: 10.1088/0951-7715/24/1/012.

[18]

C. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[19]

H. GongJ. HuangL. Liu and X. Liu, Global strong solutions of the 2D simplified Ericksen-Leslie system, Nonlinearity, 28 (2015), 3677-3694. doi: 10.1088/0951-7715/28/10/3677.

[20]

H. GongJ. Li and C. Xu, Local well-posedness of strong solutions to density-dependent liquid crystal system, Nonlinear Anal., 147 (2016), 26-44. doi: 10.1016/j.na.2016.08.014.

[21]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[22]

M. C. HongJ. Li and Z. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in R3, Comm. Partial Differential Equations, 39 (2014), 1284-1328. doi: 10.1080/03605302.2013.871026.

[23]

M. C. Hong and Z. P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in R2, Adv. Math., 231 (2012), 1364-1400. doi: 10.1016/j.aim.2012.06.009.

[24]

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.

[25]

J. HuangF. Lin and C. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in R2, Comm. Math. Phys., 331 (2014), 805-850. doi: 10.1007/s00220-014-2079-9.

[26]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[27]

F. M. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357.

[28]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[29]

J. LiE. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in R2, Math. Models Methods Appl. Sci., 26 (2016), 803-822. doi: 10.1142/S0218202516500184.

[30]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938. doi: 10.1007/s11401-010-0612-5.

[31]

F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571. doi: 10.1002/cpa.21583.

[32]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[33]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[34]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. doi: 10.3934/dcds.2011.31.1.

[35]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102.

[36]

F. H. LinJ. Y. Liu and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[37]

J. L. Lions and G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.

[38]

H. Breckner (Lisei), Approximation and optimal control of the stochastic Navier-Stokes equations, Dissertation, Martin-Luther University, Halle-Wittenberg, 1999. doi: 10.1080/02331930108844518.

[39]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012.

[40]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014.

[41]

T. Tachim Medjo, On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model, J. Differential Equations, 262 (2017), 1028-1054. doi: 10.1016/j.jde.2017.03.008.

[42]

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

[43]

E. Motyl, Martingale solution to the 2D and 3D stochastic Navier-Stokes equations driven by the compensated poisson random measure, Department of Mathematics and Computer Sciences, Lodz University, Preprint 13, 2011. doi: 10.1007/s11118-012-9300-2.

[44]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Analysis, 38 (2013), 863-912. doi: 10.1007/s11118-012-9300-2.

[45]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains- Abstract framework and applications, Stochastic Processes and their Applications, 124 (2014), 2052-2097. doi: 10.1016/j.spa.2014.01.009.

[46]

E. Pardoux, Equations and Dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse Université Paris XI, 1975.

[47]

S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, An evolution equation approach, in Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[48]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 152, Cambridge University Press, Cambridge, 2 edition, 2014. doi: 10.1017/CBO9781107295513.

[49]

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

[50]

H. Sun and C. Liu, On energetic variational approaches in modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455.

[51]

R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[52]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. doi: 10.1090/chel/343.

[53]

R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, In S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, Vol. Ⅲ, pages 535–658. Elsevier, 2004.

[54]

M. Wang and W. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962. doi: 10.1007/s00526-013-0700-y.

[55]

M. WangW. Wang and Z. Zhang, On the uniqueness of weak solution for the 2-D Ericksen-Leslie system, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 919-941. doi: 10.3934/dcdsb.2016.21.919.

[56]

H. WuX. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.

[57]

H. WuX. Xu and C. Liu, On the general ericksen-leslie system: Parodi's relation, well-posedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107. doi: 10.1007/s00205-012-0588-2.

[58]

T. Xu and T. Zhang, Large deviation principles for 2D stochastic Navier-Stokes equations driven by lévy processes, Journal of Functional Analysis, 257 (2009), 1519-1545. doi: 10.1016/j.jfa.2009.05.007.

[59]

X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181. doi: 10.1016/j.jde.2011.08.028.

[60]

W. V. LiZ. Dong and J. Zhai, Stationary weak solutions for stochastic 3d navier-stokes equations with lévy noise, Stochastic and Dynamics, 12 (2012), 1150006. doi: 10.1142/S0219493712003559.

[61]

J. Zhai and T. Zhang, Large deviations for 2D stochastic Navier-Stokes equations driven by multiplicative lévy noises, Bernoulli, 21 (2015), 2351-239. doi: 10.3150/14-BEJ647.

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