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We study the intermediate long wave equation (ILW) in negative Sobolevspaces. In particular, despite the lack of scaling invariance, we identify theregularity $s = -\frac 12$ as the critical regularity for ILW with any depthparameter, by establishing the following two results. (i) By viewing ILW as aperturbation of the Benjamin-Ono equation (BO) and exploiting the completeintegrability of BO, we establish a global-in-time a priori bound on the$H^s$-norm of a solution to ILW for $ - \frac 12 < s < 0$. (ii) By making useof explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s < - \frac12$. Our results apply to both the real line case and the periodic case.

Intermediate long wave equation in negative Sobolev spaces