On a Type I Singularity Condition in Terms of the Pressure for the Euler Equations in Double-struck capital R-3

Abstract

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions u epsilon C([0, T); W-2,W-q(R-3)), q > 3 of the incompressible Euler equations. We show that a blow up at t = T happens only if integral(T)(0) integral(t)(0) {integral(s)(0) parallel to D-2 p(tau)parallel to(L infinity) d tau exp (integral(t)(0) integral(sigma)(0) parallel to D(2)p(tau)parallel to parallel to(L infinity) d tau d sigma)} dsdt = +infinity. As consequences of this criterion we show that there is no blow up at t = T if parallel to D(2)p(t)L-infinity 8 <= c/(T-t)(2) with c 1 as t NE arrow T. Under the additional assumption of integral(T)(0) parallel to u(t)parallel to(L infinity(B(x0,rho)))dt < +infinity, we obtain localized versions of these results.