LAUSR

AbstractWe prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $${v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{{\rm loc}} (-1,0; W^{1, \infty} (B(x_0, r)))}$$v∈L∞(-1,0;L2(B(x0,r)))∩Lloc∞(-1,0;W1,∞(B(x0,r))) of the 3D Euler equations, where $${B(x_0,r)}$$B(x0,r) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as $${t\to 0}$$t→0 are small enough, then $${ \nabla v(\cdot,t) }$$∇v(·,t) does not blow-up at t = 0 in B(x0, r).