This presentation concerns questions of well-posedness and ill-posedness for the incompressible Euler equations in the plane, $\mathbb{R}^2$. We are concerned specifically with the well-known dichotomy in the Sobolev spaces, $W^{s,p}(\mathbb{R}^2)$, which states that the system is globally well-posed when the indices satisfy $1 < p < \infty$ and $s > 2/p + 1$ and are strongly ill-posed, in the sense of the nonexistence of a solution evolving continuously in $W^{s,p}(\mathbb{R}^2)$, when $1 < p < \infty$ and $s \leq 2/p + 1 $. This stark dichotomy motivates a closer investigation of the so-called critical regime where $s=2/p+1$.

We begin by exploring the borderline case where $p=1$ and consider the situation where the vorticity, which is defined as the curl of the velocity, lives in $W^{2,1}(\mathbb{R}^2)$. We show the equation for the vorticity is locally well-posed in this space. Further, with an additional assumption placed on the modulus of continuity of the initial vorticity, we show global well-posedness.

Next, we seek to refine the threshold of the well-/ill-posedness dichotomy by considering slightly modified Sobolev spaces by redefining the norm at the level of the derivative. These spaces lie strictly between the critical Sobolev spaces and the \textit{supercritical} Sobolev spaces, $W^{s,p}(\mathbb{R}^2)$ with $s > 2/p + 1$. For a particular refinement involving the use of logarithmic derivatives, we establish ill-posedness by constructing an initial velocity field where the corresponding solution immediately blows up in this space. Our strategy follows the method of large Lagrangian deformation, initially developed in the seminal work of Bourgain and Li in 2015 for the critical space $W^{2/p + 1 , p}(\mathbb{R}^2)$, $1 < p < \infty$.