On a Type I Singularity Condition in Terms of the Pressure for the Euler Equations in ℝ³

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions u∈ C([0, T); W2,q ℝ3)), q > 3 of the incompressible Euler equations. We show that a blow up at t = T happens only if 'Equation Presented' As consequences of this criterion we show that there is no blow up at t=T if ||D2 p(t)||L∞ ≤ c/(T-t)2 with c < 1 as t ↗ T. Under the additional assumption of ∫0T||u(t)||L∞ (B(x0, ρ)) dt < +∞, we obtain localized versions of these results.