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

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  • 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

    Mathematics Subject Classification: Primary: 35R60, 35Q35, 60H15, 86A05; Secondary: 76M35.


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  • [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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