Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R3 and a time interval [0,T), 0 < T ≤ ∞. It is well-known that there exists at least one global weak solution u with vanishing boundary values u∂Ω = 0 for any given initial value u0 ∈ L2σ(Ω) external force f = div F, F ∈ L2 (0,T;L2(Ω)), 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 values u∂Ω = g ∈ W 1/2,2(∂Ω). 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.
ASJC Scopus subject areas