On a Type I singularity condition in terms of the pressure for the Euler equations in \mathbb R³

2020/12/22 by Chae, Dongho, Constantin, Peter
#35Q31 #76B03 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.11948

Abstract

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions u∈ C([0, T); W^2,q (\mathbb R³)), q>3 of the incompressible Euler equations. We show that a blow up at t=T happens only if ∫₀ ^T ∫₀ ^t \∫₀ ^s ‖D² p (τ)‖_L^∞ dτexp ( ∫_s ^t ∫₀ ^\s ‖D² p (τ)‖_L^∞ dτd\s ) \dsdt = +∞. As consequences of this criterion we show that there is no blow up at t=T if ‖D² p(t)‖_L^∞ ≤ \frac c(T-t)² with c<1 as t\nearrow T. Under the additional assumption of ∫₀ ^T ‖u(t)‖_{L^∞ (B(x0, ρ))} dt

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