Trajectory attractors for the 2D Navier-Stokes system and some generalizations

Abstract

(1) ∂tu+ νLu+B(u) = g(x, t), (∇, u) = 0, u|∂Ω = 0, x ∈ Ω b R, t ≥ 0, u = u(x, t) = (u, u) ≡ u(t), g = g(x, t) = (g, g) ≡ g(t). Here Lu = −P∆u is the Stokes operator, ν > 0, B(u) = P ∑2 i=1 ui∂xiu; P is the orthogonal projector onto the space of divergence-free vector fields (see Section 1). Consider the autonomous case: g(x, t) ≡ g(x), g ∈ H, to begin with. Suppose for t = 0 we are given the initial condition