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)
On global attractors of the 3D Navier–Stokes equations. (English) Zbl 1113.35140

For certain dissipative partial differential equations unique or regular solutions in a given state space and for all time are not available, either because solutions cease to exist in the relevant topology or because it is not known whether such solutions exist in the first place. A famous example is furnished by strong solutions of the three-dimensional Navier-Stokes equations. In this situation the concept of a global attractor, the minimal closed set in the state space that uniformly attracts (in the topology of the state space) the trajectories starting from any bounded subset of initial states, is not suitable to describe the asymptotic behavior of solutions.

The work under review introduces an abstract framework which allows studying the long-term dynamics of possibly multi-valued evolution systems with respect to two related metric topologies of the state space, referred to as the weak and strong topology. While in this context the dissipative system always possesses a global attractor with respect to the weak topology (“weak global attractor”), this need not hold true in the strong topology (“strong global attractor”). However, if the strong global attractor exists and is closed in the weak topology, then it coincides with the weak global attractor.

The authors give a sufficient condition for the existence of the strong global attractor and demonstrate that this condition is satisfied for the three-dimensional space-periodic Navier-Stokes equations when all solutions on the weak global attractor are continuous in the strong topology. They extend their analysis to so-called tridiagonal models for the Navier-Stokes equations: a two-parameter family of simple models for the Navier-Stokes equations with a similar nonlinearity. These model equations fall within the framework introduced here and have weak global attractors. Solutions for certain parameter values, however, are shown to blow up in finite time (with respect to the appropriate norm). Some open questions on blow-up of solutions and strong global attractors for the tridiagonal models of the Navier-Stokes are posed and corresponding results for tridiagonal models of the Euler equations are discussed.


MSC:
35Q30Stokes and Navier-Stokes equations
35B41Attractors (PDE)