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

Abstract

First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\Vert p{\Vert }_{L_t^{\infty }(0,T^{\ast };L^{\frac{3}{2},\infty }({\mathbb{R}}^3))}\le M^2$, 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 $|v|$ and the utilization of a `small exponent'. As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows up at $T^{\ast }$ and satisfies the Type I condition $\Vert v{\Vert }_{L_t^{\infty }(0,T^{\ast };L^{3,\infty }({\mathbb{R}}^3))}\le M$, then $$\nabla v\in L^{2+O(\frac{1}{M})}({\mathbb{R}}^3\times (\frac{1}{2}{T}^{\ast },T^{\ast })).$$ 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 $v:{\mathbb{R}}^3\times [-1,0]\to {\mathbb{R}}^3$ is a weak Leray-Hopf solution to the Navier-Stokes equations with $s_n\uparrow 0$ such that $$\underset{n}{\sup }\Vert v(\cdot ,s_n){\Vert }_{L^{3,\infty }({\mathbb{R}}^3)}\le M$$ then $v$ possesses at most $O(M^{20})$ singular points at $t=0$. Our method is direct and concise. It is based upon known $\epsilon$-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.

arXiv comment15 pages. Typos with references corrected