| Abstract: |
| We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier--Stokes equations.
We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ without any smallness assumption on that scale, where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space.
As an application, we conclude that there are at most a finite number of blowup points at any singular time $t$.
The condition that the weak Lebesgue space norm of the veclocity field $u$ is bounded in time is encompassing type I singularity and significantly weaker than the end point case of the so-called Ladyzhenskaya--Prodi--Serrin condition proved by Escauriaza--Sergin--\v{S}ver\`{a}k. |
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