A critical functional framework for the inhomogeneous Navier–Stokes equations in the half-space

This paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous Navier–Stokes equations in the half-space in the case of small data with critical regularity. In dimension n 3, we state that if the initial density ρ0 is close to a positive constant in L∞ ∩ ˙W 1 n (Rn +) and the initial velocity u0 is small with respect to the viscosity in the homogeneous Besov space ˙B0 n,1(Rn +) then the equations have a unique global solution. The proof strongly relies on new maximal regularity estimates for the Stokes system in the half-space in L1(0,T ; ˙B 0 p,1(Rn +)), interesting for their own sake. © 2008 Elsevier Inc. All rights reserved