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)
Trajectory and global attractors of the three-dimensional Navier-Stokes system. (English) Zbl 1130.37404
Summary: We construct the trajectory attractor 𝔄 of a three-dimensional Navier-Stokes system with exciting force g(x)H. The set 𝔄 consists of a class of solutions to this system which are bounded in H, defined on the positive semi-infinite interval + of the time axis, and can be extended to the entire time axis so that they still remain bounded-in-H solutions of the Navier-Stokes system. In this case any family of bounded-in-L ( + ;H) solutions of this system comes arbitrary close to the trajectory attractor 𝔄. We prove that the solutions {u(x,t),t0}𝔄 are continuous in t if they are treated in the space of functions ranging in H -δ , 0<δ1. The restriction of the trajectory attractor 𝔄 to t=0, 𝔄| t=0 =:𝒜, is called the global attractor of the Navier-Stokes system. We prove that the global attractor 𝒜 thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as m the trajectory attractors 𝔄 m and the global attractors 𝒜 m of the m-order Galerkin approximations of the Navier-Stokes system converge to the trajectory and global attractors 𝔄 and 𝒜, respectively. Similar problems are studied for the cases of an exciting force of the form g=g(x,t) depending on time t and of an external force g rapidly oscillating with respect to the spatial variables or with respect to time t.
MSC:
37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
76D05Navier-Stokes equations (fluid dynamics)