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

Abstract

Consider a smooth bounded domain Ω ⊆ R with boundary ∂Ω, a time interval [0, T ), 0 < T ≤ ∞, and the Navier-Stokes system in [0, T )×Ω, with initial value u0 ∈ Lσ(Ω) and external force f = divF , F ∈ L(0, T ;L(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying |∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data |∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = v + 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. MSC: 35Q30; 35J65; 76D05