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

Abstract

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 ρ 0 is close to a positive constant in L ∞ ∩ W ˙ n 1 ( R + n ) and the initial velocity u 0 is small with respect to the viscosity in the homogeneous Besov space B ˙ n , 1 0 ( R + n ) 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 L 1 ( 0 , T ; B ˙ p , 1 0 ( R + n ) ) , interesting for their own sake.