Global well-posedness for the ILW equation in $H^s(\mathbb{T})$ for $s>-\frac12$

We prove that the intermediate long wave (ILW) equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{T})$ for $s > -\frac12$. The previous record for well-posedness was $s\geq 0$, and the system is known to be ill-posed for $s<-\frac12$. We then demonstrate that the solutions of ILW converge to those of the Benjamin--Ono equation in $H^s(\mathbb{T})$ in the infinite-depth limit. Our methods do not rely on the complete integrability of ILW, but rather treat ILW as a perturbation of the Benjamin--Ono equation by a linear term of order zero. To highlight this, we establish a general well-posedness result for such perturbations, which also applies to the Smith equation for continental-shelf waves.