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

Abstract

Abstract For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. 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 $H^1(\mathbb{R}^2)$. 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.