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 $u\in L_t^{\mathrm{\infty }}({\dot{B}}_{p,\mathrm{\infty }}^{-1+\frac{3}{p}})$ such that $\mathrm{\mid }D{\mathrm{\mid }}^{-1+\frac{3}{p}}\mathrm{\mid }u\mathrm{\mid }\in L_t^{\mathrm{\infty }}(L^p)$ with $3<p<\mathrm{\infty }$. 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 $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{B}}_{p,\mathrm{\infty }}^{-1+\frac{3}{p}}}$ combined with a single exponential of $\parallel \mathrm{\mid }D{\mathrm{\mid }}^{-1+\frac{3}{p}}\mathrm{\mid }u(t)\mathrm{\mid }{\parallel }_{L^p}$. Consequently, we obtain a new blowup rate which can be interpreted as a coupling of triple logarithm of $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{B}}_{p,\mathrm{\infty }}^{-1+\frac{3}{p}}}$ and a single logarithm of $\parallel \mathrm{\mid }D{\mathrm{\mid }}^{-1+\frac{3}{p}}\mathrm{\mid }u(t)\mathrm{\mid }{\parallel }_{L^p}$.