A minimum critical blowup rate for the high-dimensional Navier-Stokes equations

TL;DR

This paper establishes quantitative regularity and blowup criteria for high-dimensional Navier-Stokes equations, revealing a minimal growth rate of the critical norm near potential singularities, extending previous three-dimensional results.

Contribution

It provides explicit bounds on solution growth in critical spaces for dimensions d≥4, extending Tao's 3D results and quantifying blowup behavior.

Findings

Solutions in critical space grow at least as fast as a triple logarithmic rate near blowup

Explicit subcritical bounds are derived in terms of the critical norm

The results extend previous work from three to higher dimensions

Abstract

We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are obtained in terms of the critical norm. A consequence is that $\|u(t)\|_{L_x^d(\mathbb R^d)}$ grows at a minimum rate of $(\log\log\log\log(T_*-t)^{-1})^c$ along a sequence of times approaching a hypothetical blowup at $T_*$. These results quantify a theorem of Dong and Du and extend the three-dimensional work of Tao.