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

2022/01/24 by Wang, Feng-Yu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2201.09480

Abstract

For any divergence free initial datum u₀ with ‖u₀‖_∞+‖∇ u₀‖_L^p+‖∇² u₀‖_L^p<∞ for some p>d (d≥ 2), the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on ℝ^d or \mathbbT^d:=ℝ^d/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.

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