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

Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R 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 ∈ Lσ(Ω), external force f = divF , F ∈ L ( 0, T ;L(Ω) ) , and satisfying the strong energy inequality. In this paper we extend this existence result to the case of inhomogeneous boundary values u ∣∣ ∂Ω = g 6= 0. Given f as above and u0 ∈ L(Ω) satisfying the necessary compatibility conditions div u0 = 0 and N ·u0 ∣∣ ∂Ω = N ·g, where N denotes the exterior normal vector on ∂Ω, we prove as a main result the existence of a weak solution u satisfying u ∣∣ ∂Ω = g, the strong energy inequality and an energy estimate.