Martingale solution for stochastic active liquid crystal system

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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1998), 167-174. doi: 10.4213/tvp1769.
[27]	O. Kallenberg, Foundations of Modern Probability in Probabolity and Its Application, Springer-Verlag, New York, 1997.
[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.
[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.
[30]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, New York, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1998), 167-174. doi: 10.4213/tvp1769.
[27]	O. Kallenberg, Foundations of Modern Probability in Probabolity and Its Application, Springer-Verlag, New York, 1997.
[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.
[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.
[30]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford University Press, New York, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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