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

Abstract

For the d-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space Hs(Rd), s> sc := d/2 + 1. The borderline case s = sc 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 H1(R2). The main issues in 3D are vorticity stretching, lack of Lp 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.