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 = −1/2 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 Hs-norm of a solution to ILW for −1/2 <s<0. (ii) By making use of explicit 2 solutions, we prove that ILW is ill-posed in Hs for s<−1/2. Our results apply to both the real line case and the periodic case.
Original language | English |
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Pages (from-to) | 452-468 |
Number of pages | 17 |
Journal | Proceedings of the American Mathematical Society, Series B |
Volume | 11 |
Issue number | 1 |
DOIs | |
Publication status | Published - 12 Sept 2024 |
Externally published | Yes |
Keywords
- a priori bound
- Benjamin–Ono equation
- complete integrability
- ill-posedness
- Intermediate long wave equation