A SHARP CONDITION FOR EXISTENCE OF AN INERTIAL MANIFOLD Milan Miklavčič

1 Introduction

In recent years it has been shown that solutions of many important PDE’s approach exponentially to a flow on a smooth invariant finite dimensional manifold. See, for example, Henry (1981), Foias et al. (1989), Babin and Vishik (1986), Mallet–Paret and Sell (1988), Chow and Lu (1988), Hale (1989), Temam (1988) and references therein. The crucial part often lies in finding an invariant attracting manifold for the flow in a Hilbert space X generated by

u′ + Au = F (u)  (1)

where A is a sectorial operator (Henry (1981)) in X and F is such that for some α ∈ [0, 1), B ∈ B(X^{α}, X)

∥F (x) − F (y)∥ ≤ ∥B(x − y)∥ for all x, y ∈ X^{α};

here B(X^{α}, X) denotes the space of bounded operators from X^{α} (Henry (1981)) into X. Various conditions that ensure existence of an inertial (invariant and attracting) manifold for (1) are known, however, since the problem seems to be a fundamental one it would be nice to have optimal conditions. Here a condition that is sharp in a sense is presented.

Let λ > 0 be such that

λ + iω is in the resolvent set of A, ρ(A) , for all ω ∈ IR.

It is well known that A has an invariant subspace X_{1} which is the range of the projection associated with the spectral set in the half plane Re z < λ. X_{1} is an inertial manifold for u′ + Au = 0. If λ + iω ∈ ρ(A − B) for all ω ∈ IR the same could be said for the equation u′ + Au = Bu and one way to ensure this is by requiring that

∥B(A − λ − iω)^{−1}∥ < 1 for all ω ∈ IR  (2)

since

(A − B − λ − iω)^{−1} = (A − λ − iω)^{−1}(1 − B(A − λ − iω)^{−1})^{−1}.

In this paper it is proven that (2) is actually also sufficient for existence of an inertial manifold for the nonlinear equation (1) - no additional assumptions are needed. In spite of weaker and much simpler assumptions the exponential attractivity result presented here (Theorem 4.1) is actually stronger then the one obtained by Babin and Vishik (1986), Henry (1981), Chow and Lu (1988), Foias et al. (1989).

The paper is organized as follows. Assumptions, notation and some well known facts are presented in Section 2. Existence and some properties of the invariant manifold are derived in Section 3. In Section 4 exponential tracking is proven. Sections 3 and 4 are almost completely independent. In Section 5 it is shown how to modify assumptions so that the results of Sections 3, 4 become applicable also to hyperbolic problems. A comparison of various assumptions is made in Section 6.