×

zbMATH — the first resource for mathematics

Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. (English) Zbl 1117.60064
The paper presents large deviation principle for a family of stochastic 2D Navier-Stokes equations with a small additive diffusion term in bounded and unbounded domains. The existence and uniqueness of strong solutions are shown via local monotonicity arguments. Large deviations are established with the help of the weak convergence approach.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Budhiraja, A.; Dupuis, P., A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. math. statist., 20, 39-61, (2000) · Zbl 0994.60028
[2] Chang, M.-H., Large deviation for navier – stokes equations with small stochastic perturbation, Appl. math. comput., 76, 65-93, (1996) · Zbl 0851.76013
[3] Chow, P.-L., Some parabolic ito equations, Comm. pure appl. math., 45, 97-120, (1992) · Zbl 0739.60055
[4] P.-L. Chow, Introduction to Stochastic Partial Differential Equations, Lecture Notes, Wayne State University, 2000
[5] Capinsky, M.; Gatarek, D., Stochastic equations in Hilbert space with application to navier – stokes equations in any dimension, J. funct. anal., 126, 26-35, (1994) · Zbl 0817.60075
[6] Dembo, A.; Zeitouni, O., Large deviations techniques and applications, (2000), Springer-Verlag New York · Zbl 0972.60006
[7] De Simon, L., Un applicazione Della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. sem. mat. univ. Padova, 34, 205-223, (1964) · Zbl 0196.44803
[8] Dunford, N.; Schwartz, J., Linear operators, (1958), Interscience Publishers, John Wiley and Sons Inc.
[9] Dupuis, P.; Ellis, R.S., A weak convergence approach to the theory of large deviations, (1997), Wiley-Interscience New York · Zbl 0904.60001
[10] Fattorini, H.O., Infinite dimensional optimization and control theory, (1999), Cambridge University Press · Zbl 0931.49001
[11] Flandoli, F.; Gatarek, D., Martingale and stationary solutions for stochastic navier – stokes equations, Probab. theory related fields, 102, 367-391, (1995) · Zbl 0831.60072
[12] Flandoli, F.; Maslowski, B., Ergodicity of the 2-D navier – stokes equation under random perturbations, Commun. math. phys., 171, 119-141, (1995) · Zbl 0845.35080
[13] Fleming, W.H., A stochastic control approach to some large deviations problems, (), 52-66
[14] Galdi, G.P., An introduction to the theory of the navier – stokes equations, vols. 1 and 2, (1993), Springer-Verlag New York
[15] Kallianpur, G.; Xiong, J., Stochastic differential equations in infinite dimensional spaces, (1996), Institute Math. Stat. · Zbl 0859.60050
[16] Ladyzhenskaya, O.A., The mathematical theory of viscous incompressible flow, (1969), Gordon and Breach New York · Zbl 0184.52603
[17] Ladyzhenskaya, O.A.; Solonnikov, V.A., On the solvability of boundary value problems and initial – boundary value problems for the navier – stokes equations in regions with noncompact boundaries, Vestnik leningrad univ. math., 10, 271-279, (1977) · Zbl 0494.35078
[18] Menaldi, J.L.; Sritharan, S.S., Stochastic 2-D navier – stokes equation, Appl. math. optim., 46, 31-53, (2002) · Zbl 1016.35072
[19] Metivier, M., Stochastic partial differential equations in infinite dimensional spaces, (1988), Quaderni Scuola Normale Superiore, Pisa · Zbl 0664.60062
[20] E. Pardoux, Equations aux derivées partielles stochastiques non linéaires monotones, Etude des solutions fortes de type Itô, Thesis, Université de Paris Sud. Orsay, 1975
[21] Sohr, H., The navier – stokes equations: an elementary functional analytic approach, (2001), Birkhäuser Boston · Zbl 0983.35004
[22] Sowers, R., Large deviations for a reaction diffusion equation with non-Gaussian perturbations, Ann. probab., 20, 504-537, (1992) · Zbl 0749.60059
[23] Sritharan, S.S.; Sundar, P., The stochastic magneto-hydrodynamic system, Infinite dimensional analysis, quantum probab. related topics, 2, 241-265, (1999) · Zbl 0998.76099
[24] Stroock, D., An introduction to the theory of large deviations, (1984), Springer-Verlag, Universitext New York · Zbl 0552.60022
[25] Temam, R., Navier – stokes equations, theory and numerical analysis, (1984), North-Holland Amsterdam · Zbl 0568.35002
[26] Temam, R., Navier – stokes equations and nonlinear functional analysis, (1983), CBMS-NSF 41, SIAM Philadelphia, PA · Zbl 0522.35002
[27] von Wahl, W., The equations of navier – stokes and abstract parabolic equations, (1985), Friedr. Vieweg and Sohn Braunschweig
[28] Varadhan, S.R.S., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.