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 $BM{O}^{-1}$, while the low-frequency part is assumed to be small only in ${\dot{B}}_{\infty ,\infty }^{-1}$. Since $BM{O}^{-1}$ is strictly embedded in ${\dot{B}}_{\infty ,\infty }^{-1}$, 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.

arXiv comment14 pages