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Rare events in the Boussinesq system with fluctuating dynamical boundary conditions. (English) Zbl 1183.35280

Summary: The Boussinesq system models various phenomena in geophysical and climate dynamics. It is a coupled system of the Navier-Stokes equations and the salinity transport equation. Due to uncertainty in salinity flux on fluid boundary, this system is subject to random fluctuations on the boundary. This stochastic Boussinesq system can be transformed into a random dynamical system. Rare events, or small probability events, are investigated in the context of large deviations. A large deviations principle is established via a weak convergence approach based on a recently developed variational representation of functionals of infinite dimensional Brownian motion.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q35 PDEs in connection with fluid mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76M35 Stochastic analysis applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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[1] Brezis, H.; Mironescu, P., Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1, 387-404 (2001) · Zbl 1023.46031
[2] Brune, P.; Duan, J.; Schmalfuss, B., Random dynamics of the Boussinesq system with dynamical boundary conditions, Stoch. Anal. Appl., 27, 1096-1116 (2009) · Zbl 1179.60043
[3] Budhiraja, A.; Dupuis, P.; Maroulas, V., Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36, 1390-1420 (2008) · Zbl 1155.60024
[4] Cerrai, S.; Rockner, M., Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32, 1100-1139 (2004) · Zbl 1054.60065
[5] Chang, M. H., Large deviations for the Navier-Stokes equations with small stochastic perturbations, Appl. Math. Comput., 76, 65-93 (1996)
[6] Chenal, F.; Millet, A., Uniform large deviations for parabolic SPDEs and applications, Stochastic Process. Appl., 72, 161-186 (1997) · Zbl 0942.60056
[7] Chow, P. L., Large deviation problem for some parabolic Ito equations, Comm. Pure Appl. Math., 45, 97-120 (1992) · Zbl 0739.60055
[8] Chueshov, I.; Schmalfuss, B., Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17, 751-780 (2004) · Zbl 1150.60032
[9] Chueshov, I.; Schmalfuss, B., Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions, Discrete Contin. Dyn. Syst., 18, 315-338 (2007) · Zbl 1126.37050
[10] Constantin, P.; Foias, C., Navier-Stokes Equations (1988), University of Chicago Press: University of Chicago Press Chicago · Zbl 0687.35071
[11] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press · Zbl 0761.60052
[12] Dijkstra, H. A., Nonlinear Physical Oceanography (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0964.86003
[13] Duan, J.; Millet, A., Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119, 2052-2081 (2009) · Zbl 1163.60315
[14] Duan, J.; Gao, H.; Schmalfuss, B., Stochastic dynamics of a coupled atmosphere-ocean model, Stoch. Dyn., 2, 357-380 (2002) · Zbl 1090.86003
[15] Escher, J., Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18, 1309-1364 (1993) · Zbl 0816.35059
[16] Freidlin, M. I.; Wentzell, A. D., Reaction-diffusion equation with randomly perturbed boundary condition, Ann. Probab., 20, 963-986 (1992) · Zbl 0755.35055
[17] Kallianpur, G.; Xiong, J., Large deviations for a class of stochastic partial differential equations, Ann. Probab., 24, 320-345 (1996) · Zbl 0854.60026
[18] Kunita, H., Stochastic Flows and Stochastic Differential Equations (1990), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 0743.60052
[19] Ozgokmen, T.; Iliescu, T.; Fischer, P.; Srinivasan, A.; Duan, J., Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain, Ocean Model., 16, 106-140 (2007)
[20] Peszat, S., Large deviation estimates for stochastic evolution equations, Probab. Theory Related Fields, 98, 113-136 (1994) · Zbl 0792.60057
[21] Robinson, J. C., Infinite-Dimensional Dynamical Systems, Cambridge Texts Appl. Math. (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1026.37500
[22] Sowers, R., Large deviations for a reaction-diffusion system with non-Gaussian perturbations, Ann. Probab., 20, 504-537 (1992) · Zbl 0749.60059
[23] Sritharan, S. S.; Sundar, P., Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochastic Process. Appl., 116, 1636-1659 (2006) · Zbl 1117.60064
[24] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., vol. 68 (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0871.35001
[25] Wang, W.; Duan, J., Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions, Stoch. Anal. Appl., 27, 431-459 (2009) · Zbl 1166.60038
[26] Yang, D.; Duan, J., An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system, Stoch. Anal. Appl., 25, 613-639 (2007) · Zbl 1124.60052
[27] Zabczyk, J., On large deviations for stochastic evolution equations, (Stochastic Systems and Optimization. Stochastic Systems and Optimization, Lecture Notes in Control and Inform. Sci. (1988), Springer: Springer Berlin) · Zbl 0696.60056
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