zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
60H15Stochastic partial differential equations
WorldCat.org
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 Itô 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)
[6] Dembo, A.; Zeitouni, O.: Large deviations techniques and applications. (2000) · Zbl 0793.60030
[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) · Zbl 0084.10402
[9] Dupuis, P.; Ellis, R. S.: A weak convergence approach to the theory of large deviations. (1997) · Zbl 0904.60001
[10] Fattorini, H. O.: Infinite dimensional optimization and control theory. (1999) · 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. Springer lecture notes in math. 1119, 52-66 (1985)
[14] Galdi, G. P.: An introduction to the theory of the Navier--Stokes equations. 1 and 2 (1993) · Zbl 0806.35136
[15] Kallianpur, G.; Xiong, J.: Stochastic differential equations in infinite dimensional spaces. (1996) · Zbl 0859.60050
[16] Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow. (1969) · 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) · 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) · 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) · Zbl 0552.60022
[25] Temam, R.: Navier--Stokes equations, theory and numerical analysis. (1984) · Zbl 0568.35002
[26] Temam, R.: Navier--Stokes equations and nonlinear functional analysis. (1983) · Zbl 0522.35002
[27] Von Wahl, W.: The equations of Navier--Stokes and abstract parabolic equations. (1985)
[28] Varadhan, S. R. S.: Large deviations and its applications. CBMS-NSF series in applied mathematics 46 (1984)