A continuity property for the inverse of Mañé's projection

Zdeněk Skalák

Applications of Mathematics (1998)

  • Volume: 43, Issue: 1, page 9-21
  • ISSN: 0862-7940

Abstract

top
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 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 P X 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) x - y · ln ln 1 γ P x - P y 1 for every x , y X such that P x - P y 1 γ e e , where γ is a positive constant.

How to cite

top

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 -

References

top
  1. Hölder continuity for the inverse of Mañé’s projection, J. Math. Anal. Appl. vol. 178, 1993, pp. 22–29. (1993) MR1231724
  2. 10.1002/cpa.3160380102, Comm. Pure Appl. Math. vol. 38, 1985, pp. 1–27. (1985) MR0768102DOI10.1002/cpa.3160380102
  3. 10.1017/S0022112085000209, J. Fluid. Mech. vol. 150, 1985, pp. 427–440. (1985) MR0794051DOI10.1017/S0022112085000209
  4. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New-York, 1989. (1989) MR0966192
  5. Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dyn. Diff. Equ. vol. 1, 1989, pp. 45–73. (1989) MR1010960
  6. Attractors representing turbulent flows, Mem. Amer. Math. Soc. vol. 53, 1985, pp. 1–67. (1985) MR0776345
  7. [unknown], J. Math. Pures Appl. vol. 73, 1994, pp. 489–522. (1994) MR1300986
  8. Exponential attractors and their relevance to fluid dynamics systems, Phys. vol. D 63, 1993, pp. 350–360. (1993) MR1210011
  9. Local and global Lyapunov exponents, J.Dyn.Diff.Equ. vol. 3, 1991, pp. 133–177. (1991) MR1094726
  10. Construction of inertial manifolds by elliptic regularization, J. Differential Equations vol. 89, 1991, pp. 355–387. (1991) MR1091482
  11. Approximate inertial manifolds and effective viscosity in turbulent flows, Phys. Fluids vol. A 3, 1991, pp. 898–911. (1991) MR1205478
  12. 10.1006/jmaa.1993.1326, J. Math. Anal. Appl. vol. 178, 1993, pp. 567–583. (1993) MR1238896DOI10.1006/jmaa.1993.1326
  13. Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Mod. Numer. Anal. vol. 22, 1988, pp. 93–118. (1988) MR0934703
  14. Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. vol. 67, 1988, pp. 197–226. (1988) MR0964170
  15. Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations vol. 73, 1988, pp. 309–353. (1988) MR0943945
  16. Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Diff. Equ. vol. 1, 1989, pp. 199–244. (1989) MR1010966
  17. 10.1137/S0036141091224552, SIAM J. Math. Anal. vol. 25, 1994, pp. 1269–1302. (1994) MR1289139DOI10.1137/S0036141091224552
  18. 10.1137/0517080, SIAM J. Math. Anal. vol. 17, 1986, pp. 1139–1157. (1986) Zbl0626.35078MR0853521DOI10.1137/0517080
  19. 10.1137/S0036141092230428, SIAM J. Math. Anal. vol. 25, 1994, pp. 894–914. (1994) MR1271316DOI10.1137/S0036141092230428
  20. Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Univ. Math. J. vol. 41, 1992, pp. 925–981. (1992) Zbl0765.35034MR1206337
  21. 10.1016/0168-9274(94)00021-2, Appl. Num. Math. vol. 15, 1994, pp. 219–246. (1994) MR1298243DOI10.1016/0168-9274(94)00021-2
  22. On the dimension of the compact invariant sets of certain non-linear maps. Lecture Notes in Math. 898 (1981), Springer-Verlag, New York, pp. 230–242. MR0654892
  23. 10.1137/0726063, SIAM J. Numer. Anal. vol. 26, 1989, pp. 1139–1157. (1989) MR1014878DOI10.1137/0726063
  24. Integral Geometry and Geometric Probability, Addison-Wesley, Reading, 1976. (1976) Zbl0342.53049MR0433364
  25. Induced trajectories and approximate inertial manifolds, Math. Mod. Numer. Anal. vol. 23, 1989, pp. 541–561. (1989) Zbl0688.58036MR1014491
  26. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New-York, 1988. (1988) Zbl0662.35001MR0953967
  27. 10.1006/jfan.1993.1126, J. Funct. Anal. vol. 117, 1993, pp. 215–241. (1993) MR1240265DOI10.1006/jfan.1993.1126
  28. On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl. vol. 149, 1990, pp. 540–557. (1990) Zbl0723.35063MR1057693

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.