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 \({v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{{\rm loc}} (-1,0; W^{1, \infty} (B(x_0, r)))}\) of the 3D Euler equations, where \({B(x_0,r)}\) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as \({t\to 0}\) are small enough, then \({ \nabla v(\cdot,t) }\) does not blow-up at t = 0 in B(x0, r).
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Communicated by P. Constantin
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Chae, D., Wolf, J. On the Local Type I Conditions for the 3D Euler Equations. Arch Rational Mech Anal 230, 641–663 (2018). https://doi.org/10.1007/s00205-018-1254-0
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DOI: https://doi.org/10.1007/s00205-018-1254-0