Local Existence for the 2D Euler Equations in a Critical Sobolev Space

In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D incompressible Euler equations with vorticity in the critical Sobolev space $W^{s,p}({\mathbb{R}}^2)$ for $sp=2$ and $p\in (1,\mathrm{\infty })$. In this note, we establish short-time existence of solutions with vorticity in the critical space $W^{2,1}({\mathbb{R}}^2)$. Under the additional assumption that the initial vorticity is Dini continuous, we prove that the $W^{2,1}$-regularity of vorticity persists for all time.