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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

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

[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. Google Scholar

[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. Google Scholar

[49]

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

[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. Google Scholar

[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. Google Scholar

[52]

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

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

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

[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. Google Scholar

[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. Google Scholar

[49]

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

[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. Google Scholar

[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. Google Scholar

[52]

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

[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. Google Scholar

[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. Google Scholar

[55]

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