Large data existence of global-in-time strong solutions to the incompressible Navier-Stokes equations in high space dimensions
Xiangsheng Xu
TL;DR
This work addresses global-in-time existence of strong solutions to the incompressible Navier–Stokes equations in $\mathbb{R}^N$ for $N\ge 3$. The author proves that a suitable weak solution becomes strong whenever the initial velocity satisfies $|v^{(0)}|\in L^2 \cap L^{\infty}$ and is divergence-free, by establishing a time-uniform $L^{\infty}$ bound for a suitably scaled velocity through a De Giorgi–Moser iteration applied to $\psi=|u|^2$ with $u=M_\sigma^{-1}v$. Central to the argument are the scaling analysis, a local energy inequality,1 and Calderón–Zygmund pressure estimates via the Newtonian potential, which together control the pressure term and enable a global extension of the local strong solution. Consequently, the method yields a global-in-time strong solution in this setting, contributing to the Navier–Stokes millennium problem in the whole-space context. The analysis hinges on the entire-space domain to obtain the necessary a priori bounds and avoids local-in-space constraints that complicate traditional approaches.
Abstract
We study the existence of a strong solution to the initial value problem for the incompressible Navier-Stokes equations in the whole space. Our investigation shows that a ``suitable'' weak solution to the problem becomes a strong one whenever the initial velocity is divergence free and uniformly bounded with finite energy. Our results seem to have given a positive answer to the Navier-Stokes millennium problem proposed by the Clay Mathematical Institute.