Stationary Navier-Stokes Equations with Critically Singular External Forces: Existence and Stability Results

TL;DR

This paper proves the existence and stability of solutions to stationary Navier-Stokes equations with critically singular external forces in the largest known critical space, extending previous non-stationary results.

Contribution

It establishes the existence and stability of stationary solutions with critically singular forces in a new, larger critical space, advancing the understanding of Navier-Stokes equations.

Findings

Unique solutions exist for small singular external forces in a critical space.

Stationary solutions are stable under certain conditions.

The results extend previous non-stationary Navier-Stokes existence theorems.

Abstract

We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up to now for this kind of existence. This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with initial data in BMO$^{-1}$. The stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup.