Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}({\mathbb{R}}^d)$ for $d=2,3$. In two dimensions, we prove global propagation of the $W^{2,1}({\mathbb{R}}^2)$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}({\mathbb{R}}^3)$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d\mathrm{/}p,p}({\mathbb{R}}^d)$ for all $1<p<\mathrm{\infty }$, which were based on axisymmetric flows without swirl when $d=3$.