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

R. Farwig, H. Kozono, H. Sohr
2010 Funkcialaj Ekvacioj  
Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R 3 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 u 0 ∈ L 2 σ (Ω), external force f = div F , F ∈ L 2 0, T ; L 2 (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 = 0. Given f as above and u 0 ∈ L
more » ... above and u 0 ∈ L 2 (Ω) satisfying the necessary compatibility conditions div u 0 = 0 and N ·u 0 ∂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. MSC: 76D05; 35Q30; 35J65
doi:10.1619/fesi.53.231 fatcat:h2jkiqzn3rbqdjpzdrteej6pk4