  We prove local non blow-up theorems for the 3D incompressible Euler equations
under local Type I conditions. More specifically, for a classical solution
$v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1,
\infty} (B(x_0, r)))$ of the 3D Euler equations, where $B(x_0,r)$ is the ball
with radius $r$ and the center at $x_0$, if the limiting values of certain
scale invariant quantities for a solution $v(\cdot, t)$ as $t\to 0$ are small
enough, then $ \nabla v(\cdot,t) $ does not blow-up at $t=0$ in $B(x_0, r)$.
