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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  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 - A, 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

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