Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound

TL;DR

This paper establishes higher integrability results for weak solutions of the Navier-Stokes equations under certain pressure and velocity bounds, and provides bounds on the number of singular points at potential blow-up times.

Contribution

It introduces a concise method to prove higher integrability and bounds on singular points without relying on backward uniqueness or unique continuation techniques.

Findings

Higher integrability of solutions under pressure bounds

Improved integrability exponent of the velocity gradient at blow-up

Bound on the number of singular points at the first blow-up time

Abstract

First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}(\mathbb{R}^3))}\leq M^2$, then $v$ possesses higher integrability up to the first potential blow-up time $T^*$. Our method is concise and is based upon energy estimates applied to powers of $|v|$ and the utilization of a `small exponent'. As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows up at $T^*$ and satisfies the Type I condition $\|v\|_{L^{\infty}_{t}(0,T^*; L^{3,\infty}(\mathbb{R}^3))}\leq M$, then $$\nabla v\in L^{2+O(\frac{1}{M})}(\mathbb{R}^3\times (\tfrac{1}{2}T^*,T^*)).$$ This is the first result of its kind, improving the integrability exponent of $\nabla v$ under the Type I assumption in the three-dimensional setting. Finally, we show that if $v:\mathbb{R}^3\times [-1,0]\rightarrow \mathbb{R}^3$ is a weak Leray-Hopf solution to the Navier-Stokes equations with $s_{n}\uparrow 0$ such that $$\sup_{n}\|v(\cdot,s_{n})\|_{L^{3,\infty}(\mathbb{R}^3)}\leq M $$ then $v$ possesses at most $O(M^{20})$ singular points at $t=0$. Our method is direct and concise. It is based upon known $\varepsilon$-regularity, global bounds on a Navier-Stokes solution with initial data in $L^{3,\infty}(\mathbb{R}^3)$ and rescaling arguments. We do not require arguments based on backward uniqueness nor unique continuation results for parabolic operators.