Well-posedness, Smoothness and Blow-up for Incompressible Navier-Stokes Equations

TL;DR

This paper establishes well-posedness, smoothness, and blow-up criteria for the incompressible Navier-Stokes equations with specific initial data regularity, providing explicit lifespan bounds and analyzing finite-time singularities.

Contribution

It proves well-posedness and smoothness results for Navier-Stokes with initial data in certain bounded and Sobolev spaces, including explicit lifespan estimates and blow-up conditions.

Findings

Well-posedness is established for initial data with bounded and Sobolev norms.

Explicit bounds on the lifespan of solutions are derived based on initial data.

Finite-time blow-up is demonstrated for solutions with finite maximal existence time.

Abstract

For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ 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 $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.