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

Martingale solution for stochastic active liquid crystal system

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].

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. BreitE. 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. BreitE. FeireislM. 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. ChenA. MajumdarD. 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. ChenA. MajumdarD. 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. DarntonL. TurnerS. 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. DingJ. LinC. 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. DingC. 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. FeireislB. 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. FeireislA. 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. LinJ. 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. SanchezD. T. N. ChenS. J. DecampM. 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. WangX. 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. BreitE. 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. BreitE. FeireislM. 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. ChenA. MajumdarD. 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. ChenA. MajumdarD. 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. DarntonL. TurnerS. 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. DingJ. LinC. 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. DingC. 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. FeireislB. 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. FeireislA. 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. LinJ. 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. SanchezD. T. N. ChenS. J. DecampM. 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. WangX. 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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