Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions

Consider the Navier-Stokes equations in a smooth bounded domain Omega subset of R(3) and a time interval [0, T), 0 < T <= infinity. It is well-known that there exists at least one global weak solution u with vanishing boundary values u vertical bar(partial derivative Omega) = 0 for any given initial value u(0) is an element of L(sigma)(2)(Omega), external force f = div F. F is an element of L(2)(0, T; L(2)(Omega)), and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time "Leray-Hopf type" weak solutions u with nonzero boundary sal lies u vertical bar(partial derivative Omega) = g is an element of W(1/2.2)(partial derivative Omega). As for usual weak solutions we do not need any smallness condition on g; indeed, our generalized weak solutions u exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.