Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space

Abstract

We construct global smooth solutions to the incompressible Navier--Stokes equations in $\mathbb{R}^3$ for initial data in $L^2$ satisfying some smallness condition. The high-frequency part is assumed to be small in $BMO^{-1}$, while the low-frequency part is assumed to be small only in $\dot B^{-1}_{\infty,\infty}$. Since $BMO^{-1}$ is strictly embedded in $\dot B^{-1}_{\infty,\infty}$, our assumption is weaker than that of Koch and Tataru (2001), which we also demonstrate with an example of finite energy divergence-free initial data. Also, our solutions attain the initial data in the strong $L^2$ sense, and hence satisfy the energy balance for all time.