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First, we show that if the pressure $p$ (associated to a weak Leray-Hopfsolution $v$ 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 topowers of $|v|$ and the utilization of a `small exponent'.  As a consequence, we show that if a weak Leray-Hopf solution $v$ first blowsup at $T^*$ and satisfies the Type I condition $\|v\|_{L^{\infty}_{t}(0,T^*;L^{3,\infty}(\mathbb{R}^3))}\leq M$, then $$\nabla v\inL^{2+O(\frac{1}{M})}(\mathbb{R}^3\times (\tfrac{1}{2}T^*,T^*)).$$ This is thefirst 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 directand concise. It is based upon known $\varepsilon$-regularity, global bounds ona Navier-Stokes solution with initial data in $L^{3,\infty}(\mathbb{R}^3)$ andrescaling arguments. We do not require arguments based on backward uniquenessnor unique continuation results for parabolic operators.

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