Approximation of evolution differential equations in scales of Hilbert spaces

Abstract

Evolution differential equations in Hilbert spaces arise in numerous problems in different fields of natural sciences. In these problems, it is necessary to approximate infinite-dimensional systems by finite-dimensional equations (or by systems of ordinary differential equations). In the approximation of the evolution equations in scales of Hilbert spaces, high efficiency can be attained by using computers. In the present paper, we consider the problem of estimating the time of existence of solutions and approximations in the chosen scales of spaces. The results obtained allow us to develop constructive methods for obtaining such estimates in specific cases. We note that the problem of estimating the solution existence time in scales of spaces is actual both in the case of linear equations, where global solvability is possible, and in the case of nonlinear systems, where the existence time of the solution is finite. Let H be a Hilbert space. Throughout the paper, the Hilbert spaces are assumed to be separable and infinite-dimensional. We let ek denote an orthonormal basis in H. We introduce a continuous generally nonlinear operator A : D → H ,w hereD is a Hilbert space densely and continuously embedded in H and satisfying the condition {ek }⊂ D. We consider the Cauchy problem u � (t )= Au(t) ,t ∈ (0 ,T ) (1) u(0) = ϕ, ϕ ∈ H.