Effective finite parametrization in phase spaces of parabolic equations

For evolution equations of parabolic type in a Hilbert phase space $E$, consideration is given to the problem of the effective parametrization (with a Lipschitzian estimate) of the sets $\mathcal K\subset E$ by functionals $\varphi_1,\dots,\varphi_m$ in $E^*$ or, in other words, the problem of the linear Lipschitzian embedding of $\mathcal K$ in $\mathbb R^m$. If $\mathcal A$ is the global attractor for the equation, then this kind of parametrization turns out to be equivalent to the finite dimensionality of the dynamics on $\mathcal A$. Some tests are established for the parametrization (in various metrics) of subsets in $E$ and, in particular, of manifolds $\mathcal M\subset E$ by linear functionals of different classes. We outline a range of physically significant parabolic problems with a fundamental domain $\Omega\subset\mathbb R^N$ that admit a parametrization of the elements $u(x)\in\mathcal A$ by their values $u(x_i)$ at a finite system of points $x_i\in\Omega$.