Global well-posedness for the ILW equation in H^s(\mathbbT) for s>-\frac12

2025/06/05 by Louise Gassot, Gassot, Louise, Thierry Laurens +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2506.05149

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the intermediate long wave (ILW) equation is globally well-posed in the Sobolev spaces H^s(\mathbbT) for s > -\frac12. The previous record for well-posedness was s≥ 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(\mathbbT) 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.

Citations

• A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori
• Deep-water limit of the intermediate long wave equation in L²
• Intermediate long wave equation in negative Sobolev spaces
• The lifespan of small data solutions for Intermediate Long Wave equation (ILW)
• Sharp well-posedness for the Benjamin--Ono equation
• On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on \mathbbT
• On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications
• Long time behavior of solutions for a damped Benjamin-Ono equation
• On a damped Szego equation (with an appendix in collaboration with Christian Klein)
• On the integrability of the Benjamin-Ono equation on the torus
• Nonlinear wave interactions for the Benjamin-Ono equation
• Nonlocal models for nonlinear, dispersive waves

Related

• Deep-water limit of the intermediate long wave equation in L²
• Intermediate long wave equation in negative Sobolev spaces
• The lifespan of small data solutions for Intermediate Long Wave equation (ILW)
• Benjamin-Ono and Intermediate Long Wave equation : modeling, IST and PDE
• Shallow-water convergence of the intermediate long wave equation in L²