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

TL;DR

This paper proves short-time existence of solutions to the 2D Euler equations with vorticity in a critical Sobolev space and shows persistence of regularity under Dini continuity, advancing understanding of well-posedness in critical spaces.

Contribution

It establishes short-time existence in the critical space $W^{2,1}$ and persistence of regularity for initial data with Dini continuous vorticity, filling a gap in the well-posedness theory.

Findings

Existence of solutions in $W^{2,1}$ for short time.

Persistence of $W^{2,1}$-regularity under Dini continuity.

Extension of well-posedness results in critical Sobolev spaces.

Abstract

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 $p\in(1,\infty)$. 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.