The Dynamical System Generated by the 2d Navier-stokes Equations

Herein I define the global attractor for the semidynamical system (H, {S(t)}t≥0), where H is a Hilbert space and S(t) is a semigroup. In particular, I will consider the semigroup S(t) which acts on a relevant function space by S(t)u0 = u(t; u0), where u(t; u0) is the solution of a given partial differential equation at time t with initial condition u(0) = u0. By imposing regularity properties on the terms of the Navier-Stokes Equations (with periodic boundary conditions), we can find in H a maximal compact invariant set A that is also the minimal set that attracts all bounded sets X ⊂ H, and we call this set the ’global attractor’. On the global attractor, we can extend our semidynamical system to a true dynamical system (A, {S(t)}t∈R). Finally, I will show that A is finite-dimensional by constructing an explicit bound on its fractal dimension, and I will discuss in what sense this implies that the dynamics of the attractor are determined by a finite number of degrees of freedom, by showing that we can parametrize the attractor with a finite set of coordinates. The reader should have some familiarity with the languages of Banach, Hilbert and Sobolev spaces, as well as with the basic notations of PDEs and Dynamical Systems. For conciseness, some well known inequalities and estimates will be utilized without proof. This exposition most closely follows the treatment given by Robinson [11].