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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

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