A continuity property for the inverse of mañé's projection

Abstract

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d _F(X), and P ₀an orthogonal projection in H of rank greater than or equal to 2 d _F (X) + 1. For every δ > 0, there exists an orthogonal projection P in H of the same rank as P ₀, which is injective when restricted to X and such that ‖P − P ₀‖ < δ This result follows from Mañé's paper. Thus the inverse (P|_X)⁻¹ of the restricted mapping P|_X: X → PX is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé's projection (P| _X )⁻¹. It is known that when H is finite dimensional then (P| _X )⁻¹ is Hölder continuous. In this paper we shall prove that if X is a global attractor of an infinite dimensional dissipative evolutionary equation then in some cases (e.g. two-dimensional Navier-Stokes equations with homogeneous Dirichlet boundary conditions) $\parallel {\kern 1pt} x - y{\kern 1pt} \parallel \cdot \ln \ln \frac{1}{{{\gamma }\parallel Px - Py\parallel }}\;\; \leqslant \;\;1$ for every x, y ∈ X such that $\parallel Px - Py\parallel \;\; \leqslant \;\;\tfrac{1}{{{\gamma e}^{e} }}$, where γ is a positive constant.