We analyze the forced incompressible stationary Navier-Stokes flow in $\mathbb{ R}^n_+$, $n>2$. Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in homogenous Sobolev space with $s=-\frac{ 1}{ 2}$ degree of smoothness. Moreover, the velocity field is shown to be locally H\"{ o}lder continuous while the pressure belongs to $L^p_{ loc}$ for any $p\in (1,\infty)$. Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.
Categories: math.AP