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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 |
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 [
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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