Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

Abstract

We analyze the forced incompressible stationary Navier-Stokes flow in R+n, n>2. Existence of a unique solution satisfying a global integrability property measured in a scale of tent spaces is established for small data in homogeneous Sobolev space with s=−12 degree of smoothness. The velocity field is shown to be locally Hölder continuous while the pressure belongs to Llocp for any p∈(1,∞). 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.