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Effective finite parametrization in phase spaces of parabolic equations

For evolution equations of parabolic type in a Hilbert phase space , consideration is given to the problem of the effective parametrization (with a Lipschitzian estimate) of the sets by functionals in or, in other words, the problem of the linear Lipschitzian embedding of in . If 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 . Some tests are established for the parametrization (in various metrics) of subsets in and, in particular, of manifolds by linear functionals of different classes. We outline a range of physically significant parabolic problems with a fundamental domain that admit a parametrization of the elements by their values at a finite system of points .