A continuity property for the inverse of Mañé's projection

Skalák, Zdeněk. "A continuity property for the inverse of Mañé's projection." Applications of Mathematics 43.1 (1998): 9-21. <http://eudml.org/doc/32994>.

@article{Skalák1998,
abstract = {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.},
author = {Skalák, Zdeněk},
journal = {Applications of Mathematics},
keywords = {dissipative evolutionary equations; Navier-Stokes equations; attractors; Mañé’s projection; fractal dimension; dissipative evolution equations; attractors; Mañé’s projection; fractal dimension; Navier-Stokes equations},
language = {eng},
number = {1},
pages = {9-21},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A continuity property for the inverse of Mañé's projection},
url = {http://eudml.org/doc/32994},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Skalák, Zdeněk
TI - A continuity property for the inverse of Mañé's projection
JO - Applications of Mathematics
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 43
IS - 1
SP - 9
EP - 21
AB - 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.
LA - eng
KW - dissipative evolutionary equations; Navier-Stokes equations; attractors; Mañé’s projection; fractal dimension; dissipative evolution equations; attractors; Mañé’s projection; fractal dimension; Navier-Stokes equations
UR - http://eudml.org/doc/32994
ER -