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

First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\mathrm{\parallel }p{\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}(0,T^{\ast };L^{\frac{3}{2},\mathrm{\infty }}({\mathbb{R}}^3))}\le M^2$, then $v$ possesses higher integrability up to the first potential blow-up time $T^{\ast }$. Our method is concise and is based upon energy estimates applied to powers of $\mathrm{\mid }v\mathrm{\mid }$ and the utilization of a `small exponent’. As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows up at $T^{\ast }$ 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$v$possessesatmost$O(M^{20})$singularpointsat$t=0$. Our method is direct and concise. It is based upon known $\epsilon$-regularity, global bounds on a Navier-Stokes solution with initial data in $L^{3,\mathrm{\infty }}({\mathbb{R}}^3)$ and rescaling arguments. We do not require arguments based on backward uniqueness nor unique continuation results for parabolic operators.