Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space
Alexey Cheskidov, Taichi Eguchi
TL;DR
The paper proves global well-posedness for the 3D Navier--Stokes equations with small initial data lying in a frequency-localized critical space: the high-frequency part is small in $BMO^{-1}_{\sqrt{\delta}}$ while the low-frequency part is small in $\dot B^{-1}_{\infty,\infty}$. A fixed-point argument in a specially crafted scaling-invariant space $X_{T_*,\delta}$ yields a global smooth solution $u(t)$ with $u(t)\to a$ in $L^2$ as $t\to0^+$, ensuring energy balance for all times. The analysis introduces a nonlinear estimate for the localized nonlinear term, decomposing into $I_1,I_2,I_3$ and incorporating a logarithmic term via an incomplete Beta function, together with a time-scale cutoff $\delta$ to manage high-frequency interactions. The authors also construct initial data with arbitrarily large $BMO^{-1}$ norm but small $BMO^{-1}_{\sqrt{\delta}}$ and $\dot B^{-1}_{\infty,\infty}$ norms, demonstrating that their assumption is strictly weaker than Koch–Tataru. Overall, the work extends global well-posedness results to a broader class of finite-energy data and establishes energy balance from the initial time for the constructed solutions.
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.