On Rough Calderón Solutions to the Navier–Stokes Equations and Applications to the Singular Set

Abstract

In 1934, Leray (Acta Math 63:193–248, 1934) proved the existence of global-in-time weak solutions to the Navier–Stokes equations for any divergence-free initial data in \(L^2(\mathbb {R}^3)\). In the 1980s, Giga (J Differ Equ 62(2):186–212, 1986) and Kato (Math Z 187(4):471–480, 1984) independently showed that there exist global-in-time mild solutions corresponding to small enough critical \(L^3(\mathbb {R}^3)\) initial data. In 1990, Calderón (Trans Am Math Soc 318:179–200, 1990) filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in \(L^p(\mathbb {R}^3)\) for \(2< p<3\) by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a “Calderón-like” splitting to show the global-in-time existence of weak solutions to the Navier–Stokes equations corresponding to supercritical Besov space initial data \(u_0 \in \dot{B}^{s}_{{q},{\infty }}(\mathbb {R}^3)\) 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 Calderón-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.