A sufficient condition for a finite-time $ L_2 $ singularity of the 3d Euler Equation

Abstract

A sufficient condition is derived for a finite-time $L_2$ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $\lim_{t \to T_*} \sup | \frac{D \o} {Dt} |_{L_2(\vO)} = \infty$, where $~ \vO \subset \R3$ moves with the fluid. In particular, $|{\o}|$, $|\S_{ij}| , and $|\P_{ij}|$ all become unbounded at one point $(x_1,T_1)$, $T_1$ being the first blow-up time in $L_2$.