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Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
Large deviation principle for the micropolar, magneto-micropolar fluid systems
Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA |
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 [
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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