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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  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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