A continuity property for the inverse of Mañé's projection

Skalák, Zdeněk

Applications of Mathematics, Tome 43 (1998) no. 1, pp. 9-21.

Résumé

Let $X$ be a compact subset of a separable Hilbert space $H$ with finite fractal dimension $d_F(X)$, and $P_0$ an orthogonal projection in $H$ of rank greater than or equal to $2d_F(X)+1$. For every $\delta >0$, 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 $\Vert P-P_0 \Vert \delta $. This result follows from Mañé’s paper. Thus the inverse $(P \vert _X)^{-1}$ of the restricted mapping $P \vert _X\:X\rightarrow PX$ is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s projection $(P \vert _X)^{-1}$. It is known that when $H$ is finite dimensional then $(P \vert _X)^{-1}$ 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) $\Vert ~ x-y~\Vert \cdot \ln \ln \frac{1}{\gamma \Vert Px-Py \Vert }\le 1$ for every $x,y \in X$ such that $\Vert Px-Py \Vert \le \frac{1}{\gamma \mathrm{e}^{\mathrm{e}}}$, where $\gamma $ is a positive constant.

Détail

DOI : 10.1023/A:1022291923761

Classification : 35Q10, 35Q30, 37L30, 76D05, 76F99
Keywords: dissipative evolutionary equations; Navier-Stokes equations; attractors; Mañé’s projection; fractal dimension