| Title:
|
A continuity property for the inverse of Mañé's projection (English) |
| Author:
|
Skalák, Zdeněk |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
43 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
9-21 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
dissipative evolutionary equations |
| Keyword:
|
Navier-Stokes equations |
| Keyword:
|
attractors |
| Keyword:
|
Mañé’s projection |
| Keyword:
|
fractal dimension |
| MSC:
|
35Q10 |
| MSC:
|
35Q30 |
| MSC:
|
37L30 |
| MSC:
|
76D05 |
| MSC:
|
76F99 |
| idZBL:
|
Zbl 0940.35151 |
| idMR:
|
MR1488283 |
| DOI:
|
10.1023/A:1022291923761 |
| . |
| Date available:
|
2009-09-22T17:56:26Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134372 |
| . |
| Reference:
|
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