Effective finite parametrization in phase spaces of parabolic equations

Abstract

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 K ⊂ E by functionals ϕ1, . . . , ϕm in E * or, in other words, the problem of the linear Lipschitzian embedding of K in R m . If 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 A. Some tests are established for the parametrization (in various metrics) of subsets in E and, in particular, of manifolds M ⊂ E by linear functionals of different classes. We outline a range of physically significant parabolic problems with a fundamental domain Ω ⊂ R N that admit a parametrization of the elements u(x) ∈ A by their values u(xi) at a finite system of points xi ∈ Ω.