On a Type I Singularity Condition in Terms of the Pressure for the Euler Equations in ℝ3. (26th February 2021) - Chae, Dongho; Constantin, Peter
Abstract: We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions uC([0, T); W^2, q (R^3)), q>3 of the incompressible Euler equations. We show that a blow up at t=T happens only if align* &_0 ^T _0 ^t {_0 ^s \|D^2 p (τ)\|_L^∞ dτ ( _s ^t _0 ^σ \|D^2 p (τ)\|_L^∞ dτ dσ ) } ds dt = +∞.align* As consequences of this criterion we show that there is no blow up at t=T if \|D^2 p(t)\|_L^∞ c(T-t)^2 with c<1 as tT . Under the additional assumption of _0 ^T \|u(t)\|_L^∞ (B(x_0, ρ )) dt <+∞ , we obtain localized versions of these results.