In this paper we consider the
initial boundary value problem for the Navier-Stokes equations in several types of
unbounded three-dimensional domains Ω. We prove uniqueness within a class of
solutions, which we call “weak class H0 solutions”, whose members satisfy the
integrability conditions ∇u, Δu ∈ L2(0,T;L2(Ω)). Moreover, the solutions are shown
to depend continuously on their initial values. The results are based, primarily on
establishing a simple characterization of a certain space H0(Ω) of solenoidal
functions.