Henry Popkin | Searcharxiv

arXiv2026

Quantitative estimates for the forced Navier-Stokes equations and applications

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao's conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical $L^3$ norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.

math.AP↗

arXiv2024

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

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.

math.AP↗