Construction of inertial manifolds by elliptic regularization

In many cases an inertial manifold 2M 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)G₁(p, Φ(p)) + AΦ(p) = G₂(p, Φ(p)) (0) over H^p, where G₁ and G₂ 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) G₁,(p, Φ(P)) + AΦ(p) = G₂(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.