Local existence for the 2D Euler equations in a critical Sobolev space

Abstract

In the seminal work (Bourgain and Li, 2015), Bourgain and Li establish strong ill-posedness of the 2D incompressible Euler equations with vorticity in the critical Sobolev space $W^{s,p}\left(R^2\right)$ for $sp=2$ and $p\in (1,\infty )$. In this note, we establish short-time existence of solutions with vorticity in the critical space $W^{2,1}\left(R^2\right)$. 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.