Attractors of partial differential evolution equations and estimates of their dimension

Abstract

Estimates of the Hausdorff dimension of an attractor of a two-/KivioncirmQl Navipr-Stnkfis svstem 183 parabolic equations and parabolic systems 194 §10. Attractors of semigroups having a global Lyapunov function 202 §11. Regular attractors of semigroups having a Lyapunov function 206 References 210

A stochastic pitchfork bifurcation in a reaction-diffusion equation

“…By showing the existence of an unstable manifold near the origin, Babin & Vishik (1983) show that a similar lower bound holds, so that, in fact, there exists a constant c such that…”

“…Such bounds are generally based on nding some invariant manifolds, which must be subsets of the attractor (Babin & Vishik 1983). …”

Global attractors for multivalued random dynamical systems

“…The study of the asymptotic behaviour of the system is giving us relevant information about "the future" of the phenomenon described in the model. In this context, the concept of global attractor has become a very useful tool to describe the long-time behaviour of many important differential equations (see, among others, Ladyzhenkaya [18], Babin and Vishik [5], Hale [17], Temam [23]). Most of this theory has been successfully and deeply developed for autonomous deterministic partial differential equations.…”

The Century of Turbulence Theory: The Main Achievements and Unsolved Problems

“…And all the proposed scenarios were mostly compared with computations relating to some finite-dimensional models much simpler than the very intricate infinite-dimensional dynamic system evolving in the space of vector-functions of four variables in accordance with the NS equations (cf., e.g., paper [55] where a scenario for the onset of space-time chaos in a flow was studied on the model example of relatively simple nonlinear partial differential equation and it was shown that even in this case the transition to chaos proves to be quite complicated). Up to now even the question about the existence and properties of strange attractors in the infinite-dimensional phase spaces of real fluid flows is not answered satisfactorily enough (reach in content book [56] in fact covers only the attractor problem of two-dimensional fluid dynamics; see in this respect also the books listed in [134]). Thus, the completely new approaches to the transition-to-turbulence problem developed at the end of the 20th century generated, together with a number of interesting new results, also a great number of new unsolved problems which only confirm the popular assumption about the "insolvability of the problem of turbulence".…”

“…Quite impressive successes were achieved in the studies of the asymptotic behavior of the solutions of the NS equations and of the structure of the corresponding 'attractors' in the infinite-dimensional phase spaces of fluid flows; see, e.g., the books [56,135] where some of the results relating to this topic were considered. (Here again advances were most impressive in the case of 2D turbulence.)…”