Smooth attractors have zero ``thickness''
The authors study the problem of finding one-to-one finite dimensional orthogonal projections of the global attractors \(\mathcal A\) belonging to an infinite dimensional Hilbert space \(H\) such that the inverse mappings are Hölder continuous. It is known [see \textit{B. Hunt} and \textit{V. Kaloshin}, Regularity of embeddings of finite-dimensional fractal sets into infinite dimensional spaces, Nonlinearity 12, 1263-1275 (1999; Zbl 0932.28006)] that if the set \(\mathcal A\) has a finite fractal dimension \(d_f(\mathcal A)\) then a typical linear map \(L: H\to\mathbb R^k\) satisfies the condition \[ |u-v|\leq C|Lu-Lv|^\theta, \quad u,v\in\mathcal A, \quad \forall\;\theta\in\left(0 , \frac{k-2d_f(\mathcal A)}{k(1+\tau/2)}\right) \tag{1} \] where \(\tau=\tau(\mathcal A)\) is the so-called `thickness' of the set \(\mathcal A\) which measures how well the set can be approximated by finite dimensional hyperplanes. In the present paper the authors show that (1) holds for the typical orthogonal projection as well and verify that in the particular case when \(H=L^2(\Omega)\) and when the attractor is smooth (\(\mathcal A\subset W_2^s(\Omega)\) for every \(s>0\), where \(W^s_2\) are the usual Sobolev spaces) the `thickness' of the attractor is equal to zero: \(\tau(\mathcal A)=0\). This result shows that the global attractors of many physically relevant PDEs (including the 2D Navier-Stokes equations with smooth external forces in smooth domains) possess one-to-one finite dimensional orthogonal projections the inverses of which have the Hölder exponents arbitrarily close to one. Indeed, (1) implies that \[ \theta\sim 1-\frac{2d_f(\mathcal A)}k\quad \text{if } \tau(\mathcal A)=0 \] which is arbitrarily close to one if \(k\) is large enough.
scientific article; zbMATH DE number 1398594
scientific article