### Abstract:

DOI: 10.1134/S1064562406030203
Let H be a separable real Hilbert space. The norm in for all γ and γ0 that satisfy the relations
H is denoted by •, and the inner product, by 〈•, •〉. Let γ 3 1
C∞(H; 0, a), where a > 0, be the set of infinitely smooth
functions on [0, a] with values in H. The completion of -- -- -- (3)
γ 0 = δ0 – 2, –∞ γ ≤ 4, 0 δ0 2,
C∞(H; 0, a) with respect to the metric defined by the where δ0 is a constant and c depends on γ. Here, (2) and
inner product
a (3) are the conditions for the subordination of the non-
linear operator B(•, •) to A. For the three-dimensional
〈 x y〉 H2 = ∫ 〈 x t y t〉 dt, x t y t ∈ C∞ H; 0 a
Navier–Stokes equations, these conditions are fulfilled
3
0
--
is denote as H2 = H2[0, a]. at δ0 = 8 . Subordination conditions (2) and (3) were
picked according to the inequalities satisfied by the
Let A be a self-adjoint nonnegative operator with a nonlinear term in the Navier–Stokes equations. In fact,
completely continuous inverse, and let D(A) denote the these conditions can be chosen in a different manner.
domain of A. In the space of functions with values in H, For example, we can require that condition (3) hold
we consider the Cauchy problem 3
-- .
ut' + Au + B u u = f t , u 0 = 0, 0 t a, only for some γ rather than for all γ ∈ –∞, 4
(1) Definition 1. Problem (1) is said to be globally
where B(u, g) is a bilinear operator and f(t) is a function strongly solvable if, for any a > 0, the condition f(t) ∈
H2[0, a] implies that problem (1) has a solution u(t) in
with values in H.