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

Previous Article
Uniqueness and radial symmetry of minimizers for a nonlocal variational problem
CPAA Home
This Issue
Next Article
On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

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

T. Tachim Medjo

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

Keywords: Stochastic, nematic liquid crystal, variational solutions, strong solutions.

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

Citation: T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure and 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. Brzeźiak, W 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. Brzeźniak, E. 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. Caraballo, J. 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. Cavaterra, E. 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. Feireisl, M. 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. Feireisl, E. 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. Gong, J. Huang, L. 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. Gong, J. 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. Hong, J. 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. Huang, F. 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. Li, E. 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. Lin, J. 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. Wang, W. 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. Wu, X. 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. Wu, X. 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. Li, Z. 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. Brzeźiak, W 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. Brzeźniak, E. 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. Caraballo, J. 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. Cavaterra, E. 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. Feireisl, M. 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. Feireisl, E. 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. Gong, J. Huang, L. 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. Gong, J. 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. Hong, J. 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. Huang, F. 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. Li, E. 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. Lin, J. 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. Wang, W. 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. Wu, X. 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. Wu, X. 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. Li, Z. 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.

[1]	Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110
[2]	Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
[3]	Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[4]	Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561
[5]	M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497
[6]	Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[7]	Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691
[8]	Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[9]	Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[10]	Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360
[11]	Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001
[12]	Zhiyuan Geng, Wei Wang, Pingwen Zhang, Zhifei Zhang. Stability of half-degree point defect profiles for 2-D nematic liquid crystal. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6227-6242. doi: 10.3934/dcds.2017269
[13]	Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[14]	Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234
[15]	Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
[16]	Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations and Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007
[17]	Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[18]	Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[19]	Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[20]	Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
T. Tachim Medjo
on Google Scholar
T. Tachim Medjo

2020 Impact Factor: 1.916

Tools

Article outline

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
T. Tachim Medjo
on Google Scholar
T. Tachim Medjo

[Back to Top]