Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces

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 $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$.