The DNLS well-posedness threshold conjecture

Let $u=u(t,x)$ solve the derivative nonlinear Schrödinger equation on $\mathbb{R}$,

$$i\partial_tu+\partial_x^2u=i\partial_x(|u|^2u),\qquad u|_{t=0}=u_0.$$

For $s\in\mathbb{R}$, let $H^s(\mathbb{R})$ denote the Sobolev space of regularity $s$. DNLS well-posedness threshold conjecture. The DNLS equation is well-posed in $H^s(\mathbb{R})$ for $s\geq 0$, and ill-posed for $s<0$. The scaling of DNLS gives the critical Sobolev regularity $s_{\mathrm{crit}}=0$. The paper establishes norm inflation below the critical regularity for the gauged DNLS, while global well-posedness in $L^2(\mathbb{R})$ is known; the full well-posedness and ill-posedness statement for the original DNLS at all indicated regularities is presented as a conjecture.