On global attractors of the 3D Navier-Stokes equations
By A. Cheskidov, Ciprian Foias. 2006
97 citations recorded by the source corpus.
Abstract In view of the possibility that the 3D Navier–Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier–Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/0bbb5df0824021948edd929e786cbf3f29851b55
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In Applied Mathematics
- Ciprian G. Gal
- Maurizio Grasselli
- Alain Miranville
- Sergey Zelik
- A. Cheskidov
- Hao Wu
- Terence Tao
- Andrea Giorgini
In Computational Theory and Mathematics
- Alain Miranville
- Ciprian G. Gal
- Maurizio Grasselli
- Sergey Zelik
- Hao Wu
- A. Cheskidov
- Laurence Cherfils
- Chunyou Sun
In Control and Systems Engineering
- Alain Miranville
- Ciprian G. Gal
- Sergey Zelik
- Maurizio Grasselli
- Hao Wu
- A. Cheskidov
- Chunyou Sun
- Laurence Cherfils
Citations in the corpus
- Trajectory attractors for binary fluid mixtures in 3D (2010, background): “The existence of a weak global attractor in the three dimensional case for K = CH with g = 0 was firstly established in [2] for singular potentials, following the approach developed in [20].”
- Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains (2008, background): “in the multi-valued setting, see [11,19,23,24,50,158,159,201,208,209], and the references therein; see also [37,141], and [231] for nonautonomous systems.”
- Global Attractors of Evolutionary Systems (2006, background): “In addition, we discusa trajectory attractor for an evolutionary system and derive a condition under which the convergence to the trajectory attractor is strong.”
- Blow-up in finite time for the dyadic model of the Navier-Stokes equations (2006, methodology): “The following result was proved in [4]: Theorem 6.”