Intermediate long wave equation in negative Sobolev spaces

2023/11/14 by Chapouto, Andreia, Forlano, Justin, Li, Guopeng +2 · 3 citations
#35Q35 #37K10 #76B55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.08142

Abstract

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity s = -\frac 12 as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability 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 use of explicit solutions, we prove that ILW is ill-posed in H^s for s < - \frac 12. Our results apply to both the real line case and the periodic case.

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