Previous |  Up |  Next


dissipative evolutionary equations; Navier-Stokes equations; attractors; Mañé’s projection; fractal dimension
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.
[1] A.Ben-Artzi, A.Eden, C.Foiaş and B.Nicolaenko: Hölder continuity for the inverse of Mañé’s projection. J. Math. Anal. Appl. vol. 178, 1993, pp. 22–29. MR 1231724
[2] P.Constantin and C.Foiaş: Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations. Comm. Pure Appl. Math. vol. 38, 1985, pp. 1–27. DOI 10.1002/cpa.3160380102 | MR 0768102
[3] P.Constantin, C.Foiaş, O.P.Manley and R.Temam: Determining modes and fractal dimension of turbulent flows. J. Fluid. Mech. vol. 150, 1985, pp. 427–440. DOI 10.1017/S0022112085000209 | MR 0794051
[4] P.Constantin, C.Foiaş, B.Nicolaenko and R.Temam: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Springer-Verlag, New-York, 1989. MR 0966192
[5] P.Constantin, C.Foiaş, B.Nicolaenko and R.Temam: Spectral barriers and inertial manifolds for dissipative partial differential equations. J. Dyn. Diff. Equ. vol. 1, 1989, pp. 45–73. MR 1010960
[6] P.Constantin, C.Foiaş and R.Temam: Attractors representing turbulent flows. Mem. Amer. Math. Soc. vol. 53, 1985, pp. 1–67. MR 0776345
[7] A.Debussche and R.Temam: Convergent families of approximate inertial manifolds: J. Math. Pures Appl. vol. 73, 1994, pp. 489–522. MR 1300986
[8] A.Eden, C.Foiaş, B.Nicolaenko and Z.S.She: Exponential attractors and their relevance to fluid dynamics systems. Phys. vol. D 63, 1993, pp. 350–360. MR 1210011
[9] A.Eden, C.Foiaş and R.Temam: Local and global Lyapunov exponents. J.Dyn.Diff.Equ. vol. 3, 1991, pp. 133–177. MR 1094726
[10] E.Fabes, M.Luskin and G.R.Sell: Construction of inertial manifolds by elliptic regularization. J. Differential Equations vol. 89, 1991, pp. 355–387. MR 1091482
[11] C.Foiaş, O.P.Manley and R.Temam: Approximate inertial manifolds and effective viscosity in turbulent flows. Phys. Fluids vol. A 3, 1991, pp. 898–911. MR 1205478
[12] C.Foiaş, O.P.Manley and R.Temam: Iterated approximate inertial manifolds for Navier-Stokes equations in 2-D. J. Math. Anal. Appl. vol. 178, 1993, pp. 567–583. DOI 10.1006/jmaa.1993.1326 | MR 1238896
[13] C.Foiaş, O.Manley and R.Temam: Modelling of the interaction of small and large eddies in two dimensional turbulent flows. Math. Mod. Numer. Anal. vol. 22, 1988, pp. 93–118. MR 0934703
[14] C.Foiaş, B.Nicolaenko, G.Sell and R.Temam: Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimension. J. Math. Pures Appl. vol. 67, 1988, pp. 197–226. MR 0964170
[15] C.Foiaş, G.R.Sell and R.Temam: Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations vol. 73, 1988, pp. 309–353. MR 0943945
[16] C.Foiaş, G.R.Sell and E.S.Titi: Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dyn. Diff. Equ. vol. 1, 1989, pp. 199–244. MR 1010966
[17] C.Foiaş and R.Temam: Approximation of attractors by algebraic or analytic sets. SIAM J. Math. Anal. vol. 25, 1994, pp. 1269–1302. DOI 10.1137/S0036141091224552 | MR 1289139
[18] J.M.Ghidaglia: On the fractal dimension of attractors for viscous incompressible fluid flows. SIAM J. Math. Anal. vol. 17, 1986, pp. 1139–1157. DOI 10.1137/0517080 | MR 0853521 | Zbl 0626.35078
[19] D.A.Jones and E.S.Titi: A remark on quasi-stationary approximate inertial manifolds for the Navier-Stokes equations. SIAM J. Math. Anal. vol. 25, 1994, pp. 894–914. DOI 10.1137/S0036141092230428 | MR 1271316
[20] M.Kwak: Finite-dimensional inertial forms for the 2D Navier-Stokes equations. Indiana Univ. Math. J. vol. 41, 1992, pp. 925–981. MR 1206337 | Zbl 0765.35034
[21] J.Laminie, F.Pascal and R.Temam: Implementation and numerical analysis of the nonlinear Galerkin methods with finite elements discretization. Appl. Num. Math. vol. 15, 1994, pp. 219–246. DOI 10.1016/0168-9274(94)00021-2 | MR 1298243
[22] R.Mañé: 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. MR 0654892
[23] M.Marion and R.Temam: Nonlinear Galerkin methods. SIAM J. Numer. Anal. vol. 26, 1989, pp. 1139–1157. DOI 10.1137/0726063 | MR 1014878
[24] L.A.Santaló: Integral Geometry and Geometric Probability. Addison-Wesley, Reading, 1976. MR 0433364 | Zbl 0342.53049
[25] R.Temam: Induced trajectories and approximate inertial manifolds. Math. Mod. Numer. Anal. vol. 23, 1989, pp. 541–561. MR 1014491 | Zbl 0688.58036
[26] R.Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci. 68, Springer-Verlag, New-York, 1988. MR 0953967 | Zbl 0662.35001
[27] R.Temam and S.Wang: Inertial forms of Navier-Stokes equations on the sphere. J. Funct. Anal. vol. 117, 1993, pp. 215–241. DOI 10.1006/jfan.1993.1126 | MR 1240265
[28] E.S.Titi: On approximate inertial manifolds to the Navier-Stokes equations. J. Math. Anal. Appl. vol. 149, 1990, pp. 540–557. MR 1057693 | Zbl 0723.35063
Partner of
EuDML logo