On the local Type I conditions for the 3D Euler equations

Abstract

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^{\infty }(-1,0;L^2(B(x_0,r)))\cap L_{\mathrm{loc}}^{\infty }(-1,0;W^{1,\infty }(B(x_0,r)))$ 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 $B(x_0,r)$.

22 pages

References in corpus (1)

• Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II