On regularity of solutions to the Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$

TL;DR

This paper proves that solutions to the Navier-Stokes equation with initial data in BMO^{-1} are weak*-continuous over time and tend to zero in BMO^{-1} as time approaches infinity, enhancing understanding of solution regularity.

Contribution

It establishes weak*-continuity and decay properties of mild solutions in BMO^{-1}, a significant step in analyzing Navier-Stokes regularity with rough initial data.

Findings

Solutions are weak*-continuous in time in BMO^{-1}.

Global solutions vanish in BMO^{-1} at infinity.

Results apply to solutions in Koch--Tataru space.

Abstract

We prove that any mild solution in the Koch--Tataru space to the incompressible Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$ is weak*-continuous in time, valued in $\mathrm{BMO}^{-1}$. We also show that the global mild solution vanishes in $\mathrm{BMO}^{-1}$ at infinity in time.