Stationary Navier--Stokes equations on the half spaces in the scaling critical framework

Abstract

In this paper, we consider the inhomogeneous Dirichlet boundary value problem for the stationary Navier--Stokes equations in $n$-dimensional half spaces $\mathbb{R}^n_+= \{ x=(x',x_n)\ ;\ x' \in \mathbb{R}^{n-1}, x_n > 0 \}$ with $n \geq 3$ and prove the well-posedness in the scaling critical Besov spaces. Our approach is to regard the system as an evolution equation for the normal variable $x_n$ and reformulate it as an integral equation. Then, we achieve the goal by making use of the maximal regularity method that has developed in the context of nonstationary analysis in critical Besov spaces. Furthermore, for the case of $n \geq 4$, we find that the asymptotic profile of the solution as $x_n \to \infty$ is given by the $(n-1)$-dimensional stationary Navier--Stokes flow.