-
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
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 |
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 [
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
[Back to Top]