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- W4308171392 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 $$nabla vin L^{2+O(frac{1}{M})}(mathbb{R}^3times (tfrac{1}{2}T^*,T^*)).$$ 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 $v:mathbb{R}^3times [-1,0]rightarrow mathbb{R}^3$ 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." @default.
- W4308171392 created "2022-11-08" @default.
- W4308171392 creator A5042185226 @default.
- W4308171392 date "2021-11-29" @default.
- W4308171392 modified "2023-09-26" @default.
- W4308171392 title "Higher integrability and the number of singular points for the Navier-Stokes equations with a scale-invariant bound" @default.
- W4308171392 doi "https://doi.org/10.48550/arxiv.2111.14776" @default.
- W4308171392 hasPublicationYear "2021" @default.
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