Invariant manifolds for dissipative systems

Abstract

A method is given for a study of a nonlinear evolution equation for finding "slow" invariant manifolds. The method is studied for the evolution problem u =-Au+ [(Au,u) / (u,u)] u, u (0) = u0, where A is a linear, self-adjoint, possibly unbounded operator in a Hilbert space. Global existence and uniqueness of the solution to this problem are proven. Asymptotic behavior of the solution as t→∞ is studied. Analytic solution of the above nonlinear evolution problem is found. Conditions are given on the spectrum of A and on the initial data u0 for the trajectory u (t) not to have a strong limit in H as t→∞ and not to stay in any finite-dimensional space. In this sense the motion is chaotic. © 2009 American Institute of Physics.