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Well-posedness, Smoothness and Blow-up for Incompressible Navier-Stokes Equations (2201.09480v7)

Published 24 Jan 2022 in math.AP

Abstract: For any divergence free initial datum $u_0$ with $|u_0|\infty+|\nabla u_0|{Lp}+|\nabla2 u_0|_{Lp}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}d$ or $\mathbb{T}d:=\mathbb{R}d/\mathbb{Z}d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $Lp$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.

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