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In this paper, we consider the inhomogeneous Dirichlet boundary value problemfor 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. Ourapproach is to regard the system as an evolution equation for the normalvariable $x_n$ and reformulate it as an integral equation. Then, we achieve thegoal by making use of the maximal regularity method that has developed in thecontext of nonstationary analysis in critical Besov spaces. Furthermore, forthe 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--Stokesflow.

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