On rough Calder\'on solutions to the Navier-Stokes equations and applications to the singular set

Abstract

In 1934, Leray proved the existence of global-in-time weak solutions to the Navier-Stokes equations for any divergence-free initial data in $L^2$. In the 1980s, Giga and Kato independently showed that there exist global-in-time mild solutions corresponding to small enough critical $L^3(\mathbb{R}^3)$ initial data. In 1990, Calder\'on filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in $L^p$ for $2< p<3$ by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a "Calder\'on-like" splitting to show the global-in-time existence of weak solutions to the Navier-Stokes equations corresponding to supercritical Besov space initial data $u_0 \in \dot{B}^{s}_{q,\infty}$ where $q>2$ and $-1+\frac{2}{q}<s<\min \left(-1+\frac{3}{q},0 \right)$, which fills a similar gap between Leray and known mild solution theory in the Besov space setting. We also use the Calder\'on-like splitting to investigate the structure of the singular set under a Type-I blow-up assumption in the Besov space setting, which is considerably rougher than in previous works.