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

Abstract

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 ∈ L_σ²(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.