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

Xinyu He

paper

A sufficient condition for a finite-time $L_{2}$ singularity of the 3d Euler Equation

arXiv:math/0209323

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 $\lim_{t \to T_*} \sup | \frac{D ø} {Dt} |_{L_2(\vO)} = \infty$, where $~ \vO \subset \R3$ moves with the fluid. In particular, $∣ø∣$ , $∣ §_{ij} ∣, an d$ |¶_{ij}| $a l l b eco m e u nb o u n d e d a t o n e p o in t$ (x_1,T_1) $,$ T_1 $b e in g t h e f i r s t b l o w - u pt im e in$ L_2$.

AMS_Tex, 8 pages

A sufficient condition for a finite-time L2​ singularity of the 3d Euler Equation

Abstract

A sufficient condition for a finite-time $L_{2}$ singularity of the 3d Euler Equation