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

Abstract

In this paper, we address the quantitative regularity and blow-up criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space setting. Specifically, we consider solutions $u\in L^\infty_T(\dot{B}_{p,\infty}^{-1+\frac{3}{p}})$ such that $|D|^{-1+\frac{3}{p}}|u|\in L^\infty_t (L_x^p)$ with $3<p<\infty$. by establishing delicate regularity estimates and substantially improving the strategy in \cite{Tao_20}, we are able to overcome difficulties stemming from the low regularity of $\dot{B}_{p,\infty}^{-1+\frac{3}{p}}$ and derive quantitative bounds for solutions $u$ involving a quadruple exponential of their Besov norm and a double exponent of $L^p$ norm of $|D|^{-1+\frac{3}{p}}|u|$. As a consequence, we establish a new blow-up rate for solutions blowing up at a finite time expressed in terms of quadruple and double logarithms of the two critical norms respectively. This work provides improved insights into blow-up criteria and makes a significant contribution to the advancement of research on the Navier-Stokes equations in critical Besov spaces.

Locations

• arXiv (Cornell University) - View - PDF