Martingale solution for stochastic active liquid crystal system

Zhaoyang Qiu; Yixuan Wang

doi:10.3934/dcds.2020360

Previous Article
Liouville type theorems for fractional and higher-order fractional systems
DCDS Home
This Issue
Next Article
Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

May 2021, 41(5): 2227-2268. doi: 10.3934/dcds.2020360

Martingale solution for stochastic active liquid crystal system

Zhaoyang Qiu ^1, and Yixuan Wang ^2,,

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.	Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

^* Corresponding author: Yixuan Wang

Received March 2020 Revised September 2020 Published May 2021 Early access October 2020

Fund Project: The first author is supported by the CSC under grant No.201806160015

The global weak martingale solution is built through a four-level approximation scheme to stochastic compressible active liquid crystal system driven by multiplicative noise in a smooth bounded domain in $ \mathbb{R}^{3} $ with large initial data. The coupled structure makes the analysis challenging, and more delicate arguments are required in stochastic case compared to the deterministic one [11].

Keywords: Stochastic active liquid crystal system, global martingale solution, stochastic compactness, four-level approximation.

Mathematics Subject Classification: 35Q35, 76N10, 76A15.

Citation: Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360

References:

[1]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 233. Springer-Verlag, Berlin New York, 1976. Google Scholar
[2]	M. E. Bogovski$ \rm\check{i} $, Solution of some vector analysis problems connected with operators div and grad, Trudy Seminar SL Sobolev, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 80 (1980), 5-40. Google Scholar
[3]	W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172. Google Scholar
[4]	D. Breit and E. Feireisl, Stochastic Navier-Stokes-Fourier equations, Indiana Univ. Math. J., 69 (2020), 911-975. doi: 10.1512/iumj.2020.69.7895. Google Scholar
[5]	D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter Series in Applied and Numerical Mathematics, 3. De Gruyter, Berlin, 2018. Google Scholar
[6]	D. Breit, E. Feireisl and M. Hofmanová, Local strong solutions to the stochastic compressible Navier-Stokes system, Comm. Partial Differential Equations, 43 (2018), 313-345. doi: 10.1080/03605302.2018.1442476. Google Scholar
[7]	D. Breit, E. Feireisl, M. Hofmanová and B. Maslowski, Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces, Probab. Theory Related Fields, 174 (2019), 981-1032. doi: 10.1007/s00440-018-0875-4. Google Scholar
[8]	D. Breit and M. Hofmanová, Stochastic Navier-Stokes equations for compressible fluids, Idiana Univ. Math. J., 65 (2014), 1183-1250. doi: 10.1512/iumj.2016.65.5832. Google Scholar
[9]	Z. Brzeźniak and M. Ondreját, Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal., 253 (2007), 449-481. doi: 10.1016/j.jfa.2007.03.034. Google Scholar
[10]	G.-Q. Chen, A. Majumdar, D. Wang and R. Zhang, Global existence and regularity of solutions for active liquid crystals, J. Differential Equations, 263 (2017), 202-239. doi: 10.1016/j.jde.2017.02.035. Google Scholar
[11]	G.-Q. Chen, A. Majumdar, D. Wang and R. Zhang, Global weak solutions for the compressible active liquid crystal system, SIAM J. Math. Anal., 50 (2018), 3632-3675. doi: 10.1137/17M1156897. Google Scholar
[12]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar
[13]	N. C. Darnton, L. Turner, S. Rojevsk and H. C. Berg, Dynamics of bacterial swarming, Biophys. J., 98 (2010), 2082-2090. doi: 10.1016/j.bpj.2010.01.053. Google Scholar
[14]	S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539. Google Scholar
[15]	S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357. Google Scholar
[16]	C. R. Doering and J. D. Gibbon. Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511608803. Google Scholar
[17]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. Google Scholar
[18]	E. Feireisl, B. Maslowski and A. Novotný, Compressible fluid flows driven by stochastic forcing, J. Differential Equations, 254 (2013), 1342-1358. doi: 10.1016/j.jde.2012.10.020. Google Scholar
[19]	E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid. Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. Google Scholar
[20]	F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Math., Springer, Berlin, 1942 (2008), 51-150. doi: 10.1007/978-3-540-78493-7_2. Google Scholar
[21]	F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. Google Scholar
[22]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar
[23]	I. Gyöngy and N. Krylov, Existence of strong solutions for Itôs stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. Google Scholar
[24]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346. Google Scholar
[25]	X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar
[26]	A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1998), 167-174. doi: 10.4213/tvp1769. Google Scholar
[27]	O. Kallenberg, Foundations of Modern Probability in Probabolity and Its Application, Springer-Verlag, New York, 1997. Google Scholar
[28]	W. Lian and R. Zhang, Global weak solutions to the active hydrodynamics of liquid crystals, J. Differential Equations, 268 (2019), 4194-4221. doi: 10.1016/j.jde.2019.10.020. Google Scholar
[29]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. Google Scholar
[30]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, New York, 1996. Google Scholar
[31]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. Google Scholar
[32]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkh?user Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. Google Scholar
[33]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan. Acad. Ser. A Math. Sci., 55 (1979), 337-342. doi: 10.3792/pjaa.55.337. Google Scholar
[34]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. doi: 10.1215/kjm/1250522322. Google Scholar
[35]	A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738. Google Scholar
[36]	M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049. doi: 10.1137/10079224X. Google Scholar
[37]	M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67. doi: 10.1007/s00205-011-0443-x. Google Scholar
[38]	S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys., 1 (2010), 323-345. doi: 10.1146/annurev-conmatphys-070909-104101. Google Scholar
[39]	S. A. Smith, Random perturbations of viscous, compressible fluids: Global existence of weak solutions, SIAM J. Math. Anal., 49 (2017), 4521-4578. doi: 10.1137/15M1015340. Google Scholar
[40]	T. Sanchez, D. T. N. Chen, S. J. Decamp, M. Heymann and Z. Dogic, Spontaneous motion in hierarchically assembled active matter, Nature, 491 (2012), 431-434. doi: 10.1038/nature11591. Google Scholar
[41]	S. A. Smith and K. Trivisa, The stochastic Navier-Stokes equations for heat-conducting, compressible fluids: global existence of weak solutions, J. Evolution Equations, 18 (2018), 411-465. doi: 10.1007/s00028-017-0407-1. Google Scholar
[42]	D. Wang and H. Wang, Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations, Differential Integral Equations, 28 (2015), 1105-1154. Google Scholar
[43]	D. Wang, X. Xu and C. Yu, Global weak solution for a coupled compressible Navier-Stokes and $Q$-tensor system, Commun. Math. Sci., 13 (2015), 49-82. doi: 10.4310/CMS.2015.v13.n1.a3. Google Scholar
[44]	D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x. Google Scholar

show all references

References:

[1]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 233. Springer-Verlag, Berlin New York, 1976. Google Scholar
[2]	M. E. Bogovski$ \rm\check{i} $, Solution of some vector analysis problems connected with operators div and grad, Trudy Seminar SL Sobolev, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 80 (1980), 5-40. Google Scholar
[3]	W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172. Google Scholar
[4]	D. Breit and E. Feireisl, Stochastic Navier-Stokes-Fourier equations, Indiana Univ. Math. J., 69 (2020), 911-975. doi: 10.1512/iumj.2020.69.7895. Google Scholar
[5]	D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter Series in Applied and Numerical Mathematics, 3. De Gruyter, Berlin, 2018. Google Scholar
[6]	D. Breit, E. Feireisl and M. Hofmanová, Local strong solutions to the stochastic compressible Navier-Stokes system, Comm. Partial Differential Equations, 43 (2018), 313-345. doi: 10.1080/03605302.2018.1442476. Google Scholar
[7]	D. Breit, E. Feireisl, M. Hofmanová and B. Maslowski, Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces, Probab. Theory Related Fields, 174 (2019), 981-1032. doi: 10.1007/s00440-018-0875-4. Google Scholar
[8]	D. Breit and M. Hofmanová, Stochastic Navier-Stokes equations for compressible fluids, Idiana Univ. Math. J., 65 (2014), 1183-1250. doi: 10.1512/iumj.2016.65.5832. Google Scholar
[9]	Z. Brzeźniak and M. Ondreját, Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal., 253 (2007), 449-481. doi: 10.1016/j.jfa.2007.03.034. Google Scholar
[10]	G.-Q. Chen, A. Majumdar, D. Wang and R. Zhang, Global existence and regularity of solutions for active liquid crystals, J. Differential Equations, 263 (2017), 202-239. doi: 10.1016/j.jde.2017.02.035. Google Scholar
[11]	G.-Q. Chen, A. Majumdar, D. Wang and R. Zhang, Global weak solutions for the compressible active liquid crystal system, SIAM J. Math. Anal., 50 (2018), 3632-3675. doi: 10.1137/17M1156897. Google Scholar
[12]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar
[13]	N. C. Darnton, L. Turner, S. Rojevsk and H. C. Berg, Dynamics of bacterial swarming, Biophys. J., 98 (2010), 2082-2090. doi: 10.1016/j.bpj.2010.01.053. Google Scholar
[14]	S. Ding, J. Lin, C. Wang and H. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539. Google Scholar
[15]	S. Ding, C. Wang and H. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357. Google Scholar
[16]	C. R. Doering and J. D. Gibbon. Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511608803. Google Scholar
[17]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. Google Scholar
[18]	E. Feireisl, B. Maslowski and A. Novotný, Compressible fluid flows driven by stochastic forcing, J. Differential Equations, 254 (2013), 1342-1358. doi: 10.1016/j.jde.2012.10.020. Google Scholar
[19]	E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid. Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. Google Scholar
[20]	F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Math., Springer, Berlin, 1942 (2008), 51-150. doi: 10.1007/978-3-540-78493-7_2. Google Scholar
[21]	F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467. Google Scholar
[22]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar
[23]	I. Gyöngy and N. Krylov, Existence of strong solutions for Itôs stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158. doi: 10.1007/BF01203833. Google Scholar
[24]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346. Google Scholar
[25]	X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. Google Scholar
[26]	A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1998), 167-174. doi: 10.4213/tvp1769. Google Scholar
[27]	O. Kallenberg, Foundations of Modern Probability in Probabolity and Its Application, Springer-Verlag, New York, 1997. Google Scholar
[28]	W. Lian and R. Zhang, Global weak solutions to the active hydrodynamics of liquid crystals, J. Differential Equations, 268 (2019), 4194-4221. doi: 10.1016/j.jde.2019.10.020. Google Scholar
[29]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. Google Scholar
[30]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, New York, 1996. Google Scholar
[31]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. Google Scholar
[32]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkh?user Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. Google Scholar
[33]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan. Acad. Ser. A Math. Sci., 55 (1979), 337-342. doi: 10.3792/pjaa.55.337. Google Scholar
[34]	A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. doi: 10.1215/kjm/1250522322. Google Scholar
[35]	A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738. Google Scholar
[36]	M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049. doi: 10.1137/10079224X. Google Scholar
[37]	M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67. doi: 10.1007/s00205-011-0443-x. Google Scholar
[38]	S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys., 1 (2010), 323-345. doi: 10.1146/annurev-conmatphys-070909-104101. Google Scholar
[39]	S. A. Smith, Random perturbations of viscous, compressible fluids: Global existence of weak solutions, SIAM J. Math. Anal., 49 (2017), 4521-4578. doi: 10.1137/15M1015340. Google Scholar
[40]	T. Sanchez, D. T. N. Chen, S. J. Decamp, M. Heymann and Z. Dogic, Spontaneous motion in hierarchically assembled active matter, Nature, 491 (2012), 431-434. doi: 10.1038/nature11591. Google Scholar
[41]	S. A. Smith and K. Trivisa, The stochastic Navier-Stokes equations for heat-conducting, compressible fluids: global existence of weak solutions, J. Evolution Equations, 18 (2018), 411-465. doi: 10.1007/s00028-017-0407-1. Google Scholar
[42]	D. Wang and H. Wang, Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations, Differential Integral Equations, 28 (2015), 1105-1154. Google Scholar
[43]	D. Wang, X. Xu and C. Yu, Global weak solution for a coupled compressible Navier-Stokes and $Q$-tensor system, Commun. Math. Sci., 13 (2015), 49-82. doi: 10.4310/CMS.2015.v13.n1.a3. Google Scholar
[44]	D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x. Google Scholar

[1]	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
[2]	Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[3]	Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473
[4]	Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
[5]	Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure & Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038
[6]	Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[7]	Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193
[8]	Xiuqing Wang, Yuming Qin, Alain Miranville. Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3805-3827. doi: 10.3934/cpaa.2020168
[9]	Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
[10]	Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[11]	Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[12]	Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065
[13]	Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[14]	Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[15]	Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[16]	Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001
[17]	Renhui Wan. The 3D liquid crystal system with Cannone type initial data and large vertical velocity. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5521-5539. doi: 10.3934/dcds.2017240
[18]	Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[19]	Yuanshi Wang, Hong Wu, Shigui Ruan. Global dynamics and bifurcations in a four-dimensional replicator system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 259-271. doi: 10.3934/dcdsb.2013.18.259
[20]	Kexue Li. Effects of the noise level on nonlinear stochastic fractional heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5437-5460. doi: 10.3934/dcdsb.2019065

American Institute of Mathematical Sciences

Martingale solution for stochastic active liquid crystal system

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Martingale solution for stochastic active liquid crystal system

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors