Construction of inertial manifolds by elliptic regularization

Abstract

In many cases an inertial manifold 2 M for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function Φ from a finite dimensional Hilbert space H p to another Hilbert space H q . The invariance property of M means that Φ can be written as the solution of a first order partial differential equation DΦ( p) G 1( p, Φ( p)) + AΦ( p) = G 2( p, Φ( p)) (0) over H p , where G 1 and G 2 are nonlinear functions which depend on the original dynamical system and A is a suitably “stable” linear operator. In this paper we use a method introduced by Sacker (R. J. Sacker, A new approach to the perturbation theory of invariant surface, Comm. Pure Appl. Math. 18 (1965), 717–732), for the study of finite dimensional dynamical systems, to find inertial manifolds in the infinite dimensional setting. This method involves replacing the first order equation for Φ by the regularized elliptic equation − εΔΦ + DΦ( p) G 1,( p, Φ( P)) + AΦ( p) = G 2( p, Φ( p)), with suitable boundary conditions. It is shown that if A satisfies a spectral gap condition, then the solutions Φ ε of the elliptic equation converge to a weak solution Φ of (0), as ε → 0 +. Furthermore, M = Graph Φ is an invariant manifold for the given dynamical system.