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March  2018, 23(2): 913-938. doi: 10.3934/dcdsb.2018048

Large deviation principle for the micropolar, magneto-micropolar fluid systems

Kazuo Yamazaki

Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

Received  January 2017 Revised  August 2017 Published  March 2018 Early access  December 2017

Micropolar fluid and magneto-micropolar fluid systems are systems of equations with distinctive feature in its applicability and also mathematical difficulty. The purpose of this work is to follow the approach of [8] and show that another general class of systems of equations, that includes the two-dimensional micropolar and magneto-micropolar fluid systems, is well-posed and satisfies the Laplace principle, and consequently the large deviation principle, with the same rate function.

Keywords: Laplace principle, large deviation principle, micropolar fluid system, magneto-micropolar fluid system, well-posedness.
Mathematics Subject Classification: Primary: 35Q35, 37L55; Secondary: 60F10.
Citation: Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[3]

J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.

[4]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. 

[5]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.

[6]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139.  doi: 10.1214/aop/1079021473.

[7]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, App. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[9]

I. Chueshov and A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch. Anal. Appl., 29 (2011), 570-611.  doi: 10.1080/07362994.2011.581081.

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.

[11]

N. J. Cutland and B. Enright, Stochastic nonhomogeneous incompressible Navier-Stokes equations, J. Differential Equations, 228 (2006), 140-170.  doi: 10.1016/j.jde.2006.04.009.

[12]

G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, U. K., 2014.

[13]

A. Dembo and O. Zeitouni Large Deviations Techniques and Applications, Springer-Verlag, Berlin, Heidelberg, 1998.

[14]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.

[15]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081.  doi: 10.1016/j.spa.2008.10.004.

[16]

P. Dupuis and R. S. Ellis A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997.

[17]

A. C. Eringen, Simple microfluids, Int. J. Engng. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.

[18]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. 

[19]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[20]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, Heidelberg, New York, Dordrecht, London, 2012.

[21]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.

[22]

T. Hmidi, S. Keraani and R. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[23]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Third Edition, Springer, New York, 1991.

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, United Kingdom, 1990.

[25]

J. L. Lions, Quelquels Méthodes de Réesolution des Problémes aux Limites Non Linéaires, Dunod, Gauthiers-Villars, Paris, 1969.

[26]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkh mat$\ddot h$rmauser, Boston, 1999.

[27]

S. J. A. Malham, Regularity Assumptions and Length Scales for the Navier-Stokes Equations, Ph. D thesis, University of London, 1993.

[28]

U. Manna, S. S. Sritharan and P. Sundar, Large deviations for the stochastic shell model of turbulence, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 493-521.  doi: 10.1007/s00030-009-0023-z.

[29]

J.-L. Menaldi and S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.

[30]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.

[31]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259. 

[32]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.

[33]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.

[34]

A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl., 1 (1956), 261-290.  doi: 10.1137/1101022.

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer-Verlag New York, Inc., 1997.

[37]

K. Yamazaki, 3-D stochastic micropolar and magneto-micropolar fluid systems with non-Lipschitz multiplicative noise, Commun. Stoch. Anal., 8 (2014), 413-437. 

[38]

K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.  doi: 10.3934/dcds.2015.35.2193.

[39]

K. Yamazaki, Recent developments on the micropolar and magneto-micropolar fluid systems: Deterministic and stochastic perspectives, in Stochastic Equations for Complex Systems: Theoretical and Computational Topics (eds. S. Heinz and H. Bessaih), Springer International Publishing, (2015), 85-103.  doi: 10.1007/978-3-319-18206-3_4.

[40]

K. Yamazaki, Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system, Adv. Differential Equations, 21 (2016), 1085-1116. 

[41]

K. Yamazaki, Exponential convergence of the stochastic micropolar and magneto-micropolar fluid systems, Commun. Stoch. Anal., 10 (2016), 271-295. 

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[3]

J. L. Boldrini, M. A. Rojas-Medar and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl., 82 (2003), 1499-1525.  doi: 10.1016/j.matpur.2003.09.005.

[4]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. 

[5]

A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420.  doi: 10.1214/07-AOP362.

[6]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139.  doi: 10.1214/aop/1079021473.

[7]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, App. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[9]

I. Chueshov and A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch. Anal. Appl., 29 (2011), 570-611.  doi: 10.1080/07362994.2011.581081.

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.

[11]

N. J. Cutland and B. Enright, Stochastic nonhomogeneous incompressible Navier-Stokes equations, J. Differential Equations, 228 (2006), 140-170.  doi: 10.1016/j.jde.2006.04.009.

[12]

G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, U. K., 2014.

[13]

A. Dembo and O. Zeitouni Large Deviations Techniques and Applications, Springer-Verlag, Berlin, Heidelberg, 1998.

[14]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.

[15]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081.  doi: 10.1016/j.spa.2008.10.004.

[16]

P. Dupuis and R. S. Ellis A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997.

[17]

A. C. Eringen, Simple microfluids, Int. J. Engng. Sci., 2 (1964), 205-217.  doi: 10.1016/0020-7225(64)90005-9.

[18]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. 

[19]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Relat. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[20]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Third Edition, Springer, Heidelberg, New York, Dordrecht, London, 2012.

[21]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.

[22]

T. Hmidi, S. Keraani and R. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[23]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Third Edition, Springer, New York, 1991.

[24]

H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, United Kingdom, 1990.

[25]

J. L. Lions, Quelquels Méthodes de Réesolution des Problémes aux Limites Non Linéaires, Dunod, Gauthiers-Villars, Paris, 1969.

[26]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkh mat$\ddot h$rmauser, Boston, 1999.

[27]

S. J. A. Malham, Regularity Assumptions and Length Scales for the Navier-Stokes Equations, Ph. D thesis, University of London, 1993.

[28]

U. Manna, S. S. Sritharan and P. Sundar, Large deviations for the stochastic shell model of turbulence, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 493-521.  doi: 10.1007/s00030-009-0023-z.

[29]

J.-L. Menaldi and S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim., 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.

[30]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.

[31]

M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259. 

[32]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.

[33]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.

[34]

A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl., 1 (1956), 261-290.  doi: 10.1137/1101022.

[35]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116 (2006), 1636-1659.  doi: 10.1016/j.spa.2006.04.001.

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer-Verlag New York, Inc., 1997.

[37]

K. Yamazaki, 3-D stochastic micropolar and magneto-micropolar fluid systems with non-Lipschitz multiplicative noise, Commun. Stoch. Anal., 8 (2014), 413-437. 

[38]

K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.  doi: 10.3934/dcds.2015.35.2193.

[39]

K. Yamazaki, Recent developments on the micropolar and magneto-micropolar fluid systems: Deterministic and stochastic perspectives, in Stochastic Equations for Complex Systems: Theoretical and Computational Topics (eds. S. Heinz and H. Bessaih), Springer International Publishing, (2015), 85-103.  doi: 10.1007/978-3-319-18206-3_4.

[40]

K. Yamazaki, Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system, Adv. Differential Equations, 21 (2016), 1085-1116. 

[41]

K. Yamazaki, Exponential convergence of the stochastic micropolar and magneto-micropolar fluid systems, Commun. Stoch. Anal., 10 (2016), 271-295. 

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