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

TL;DR

This paper proves the existence, uniqueness, and regularity of steady-state solutions to the Navier-Stokes equations with irregular data, using advanced function space analysis and new solvability results for the Stokes system.

Contribution

It introduces a novel approach to establish solutions with irregular data in tent spaces and Triebel-Lizorkin classes, expanding the understanding of Navier-Stokes regularity.

Findings

Existence of unique solutions with small data in tent spaces.

Velocity field is locally Hölder continuous.

Pressure belongs to local L^p spaces for all p in (1,∞).

Abstract

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.