Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces
Ruilin Hu, Phuoc-Tai Nguyen, Quoc-Hung Nguyen, Ping Zhang
TL;DR
This work provides quantitative regularity and blow-up criteria for the 3D Navier–Stokes equations in endpoint critical Besov spaces. It extends Tao’s quantitative approach to Besov-type scaling by introducing an iteration-based decomposition $u=\sum_{k=1}^m u_{kL}+v$, achieving stepwise regularity gains in critical spaces via Kato-norms and energy estimates. The authors derive a bound of the form a nested exponential in $\|u(t)\|_{\dot{B}_{p,\infty}^{-1+3/p}}$ coupled with an exponential in $\bigl\| |D|^{-1+3/p}|u(t)|\bigr\|_{L^p}$, and obtain a blow-up rate that ties a triple logarithm of the Besov norm to a single logarithm of the weighted $L^p$ norm. They also prove a quantitative blow-up criterion: if the solution blows up in finite time, certain Besov–$L^p$ quantities must diverge at a triple-exponential scale, sharpening the understanding of endpoint regularity in critical Besov spaces.
Abstract
In this paper, we study the quantitative regularity and blowup criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space framework. Specifically, we consider solutions $u\in L^\infty_t(\dot{B}_{p,\infty}^{-1+\frac{3}{p}})$ such that $|D|^{-1+\frac{3}{p}}|u|\in L^\infty_t (L^p)$ with $3<p<\infty$. By deriving refined regularity estimates and substantially improving the strategy in \cite{Tao_20}, we overcome difficulties stemming from the low regularity of the Besov spaces and establish quantitative bounds for such solutions. These bounds are expressed in terms of a triple exponential of $\| u (t)\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ combined with a single exponential of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$. Consequently, we obtain a new blowup rate which can be interpreted as a coupling of triple logarithm of $\| u(t) \|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ and a single logarithm of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$.