turkmath.org

I will try to expose some attempts of obtaining a finite dimensional description for dissipative partial differential equations. One of the main aims of this theory is to obtain such an explicit description in the case of the two dimensional incompressible Navier-Stokes equations with periodic boundary conditions. The concepts of inertial manifolds, exponential attractors, imbedding by Hölder-Mané projections, bi-Lipschitzian imbeddibility of the attractor will be briefly discussed. I will end with a recent counterexample constructed in Eden, Kalantarov and Zelik. (Counterexamples to the regularity of Mane's Projections and Global Attractors, Russian Mathematical Survey, 68:2 (2013) 199–226)