Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

Consider a smooth bounded domain Ω ⊆ ℝ³ with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u₀ ∈ L²_σ(Ω) and external force f = div F, F ∈ L²(0, T;L²(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe}_∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe}_∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.