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)
Ergodicity for the dissipative Boussinesq equations with random forcing. (English) Zbl 1113.35141
For the two-dimensional stochastically forced Boussinesq equation the existence of a unique stationary measure is proved if all the determining modes are forced. The existence of an invariant measure is proved by the standard compactness argument. The proof of the uniqueness is reduced to two parts. In the first part the regularity of the transition probability densities is shown, i.e. the hypoellipticity of the diffusion operator. Then the authors prove the irreducibility of the associated Markov process in the sense starting from any initial position, the dynamics enter any neighborhood of the origin infinitely often. These two results allow to prove the main ergodicity result following to the previous results on the stochastically forced Navier-Stokes equations. The Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing are also studied.
MSC:
35Q35PDEs in connection with fluid mechanics
35Q60PDEs in connection with optics and electromagnetic theory
60H40White noise theory
76D99Incompressible viscous fluids
76M35Stochastic analysis (fluid mechanics)