zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains. (English) Zbl 1113.60062
This paper is devoted to study the behaviour of stochastic partial differential equations in unbounded domains. The authors begin by considering random dynamical systems (RDS) on a separable Banach space. They introduce the new concept of asymptotically compact (AC) RDS. They prove that for an ACRDS, the $\Omega$-limit set of any bounded subset $B$ is nonempty, compact, invariant and attracts $B$. The invariance of this set will imply the existence of invariant measure, while the uniqueness is not considered. Using the classical Galerkin approximation method and some compactness theorem they prove the existence of the stochastic flow associated with 2D stochastic Navier-Stokes equations in Poincaré domains (possibly unbounded). Then, they construct the RDS corresponding to the Navier-Stokes equations. Using energy inequalities they are able to prove the continuity of this RDS in a weak topology and that this RDS is AC. Finally, they obtain the existence of an invariant measure for the 2D stochastic Navier-Stokes equations perturbed by an additive noise. Since the method used by the authors does not depend on the compactness of the Sobolev embeddings, they can deal with unbounded domains and they can relax assumptions on the noise. So, they obtain new results in bounded and in unbounded domains.

60H15Stochastic partial differential equations
35R60PDEs with randomness, stochastic PDE
37H10Generation, random and stochastic difference and differential equations
34F05ODE with randomness
Full Text: DOI