Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

In an exterior domain Omega subset of R-3 and a time interval [0, T), 0 < T <= infinity, consider the instationary Navier-Stokes equations with initial value u(0) epsilon L-sigma(2)(Omega) and external force f = div F, F epsilon L-2(0, T; L-2(Omega)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray Hopf weak solutions to the case when (u)vertical bar(partial derivative Omega) = g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate. (C) 2014 Elsevier Inc. All rights reserved.