Strong Ill-Posedness of the 3D Incompressible Euler Equation in Borderline Spaces

For the d-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space H-s(R-d), s > s(c) := d/2 + 1. The borderline case s = s(c) was a folklore conjecture. In the previous paper [2], we introduced a new strategy (large lagrangian deformation and high frequency perturbation) and proved strong ill-posedness in the critical space H-1(R-2). The main issues in 3D are vorticity stretching, lack of L-p conservation, and control of lifespan. Nevertheless in this work we overcome these difficulties and show strong ill-posedness in 3D. Our results include general borderline Sobolev and Besov spaces.