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Unconditional deep-water limit of the intermediate long wave equation in low-regularity

Justin Forlano, Guopeng Li, Tengfei Zhao

TL;DR

This work proves an unconditional deep-water limit for the intermediate long wave equation (ILW) converging to the Benjamin-Ono (BO) equation in low-regularity Sobolev spaces on both the real line and the circle. The authors view ILW as a BO perturbation with a smoothing term and develop an unconditional uniqueness theory in $C(\mathbb{R};H^s)$ for ILW and BO down to $s>s_0$ with $s_0=3-\sqrt{33/4}\approx 0.1277$, thereby upgrading prior conditional results to unconditional convergence. Central to the approach are a refined gauge transform, two rounds of normal form reductions, and refined Strichartz estimates that exploit the smoothing effect of the ILW perturbation. The results hold on $\mathbb{R}$ for $s_0<s\le 1/4$ and on $\mathbb{T}$ for $s_0<s<1/2$, and extend to a two-depth ILW setting as well as to the shallow-water regime with appropriate rescaling. The work thus bridges ILW and BO in the lowest feasible regularity, with significant implications for the rigorous understanding of deep-water limits in dispersive PDEs.

Abstract

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in $H^s$ when $s_0<s\leq \frac 14$ on the line and $s_0<s< \frac 12$ on the circle, where $s_0 = 3-\sqrt{33/4}\approx 0.1277$. Here, we adapt the strategy of Moşincat-Pilod (2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.

Unconditional deep-water limit of the intermediate long wave equation in low-regularity

TL;DR

This work proves an unconditional deep-water limit for the intermediate long wave equation (ILW) converging to the Benjamin-Ono (BO) equation in low-regularity Sobolev spaces on both the real line and the circle. The authors view ILW as a BO perturbation with a smoothing term and develop an unconditional uniqueness theory in for ILW and BO down to with , thereby upgrading prior conditional results to unconditional convergence. Central to the approach are a refined gauge transform, two rounds of normal form reductions, and refined Strichartz estimates that exploit the smoothing effect of the ILW perturbation. The results hold on for and on for , and extend to a two-depth ILW setting as well as to the shallow-water regime with appropriate rescaling. The work thus bridges ILW and BO in the lowest feasible regularity, with significant implications for the rigorous understanding of deep-water limits in dispersive PDEs.

Abstract

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in when on the line and on the circle, where . Here, we adapt the strategy of Moşincat-Pilod (2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.
Paper Structure (10 sections, 20 theorems, 123 equations)

This paper contains 10 sections, 20 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Gauge transform and Strichartz estimates
  3. Notation and preliminary estimates
  4. Gauge transformation on $\mathbb{R}$
  5. Strichartz estimates
  6. Normal form reductions on $\mathbb{R}$
  7. Preparations for normal forms
  8. First step of normal form reductions
  9. Second step of the normal form reductions
  10. On the periodic setting

Key Result

Theorem 1.1

Let $\mathcal{M} =\mathbb{R}$ or $\mathbb{T}$ and $s_0<s<\frac{1}{2}$, where $s_0 := 3-\sqrt{33/4}\approx 0.1277$. Given $u_0\in H^{s}(\mathcal{M})$, let $\{u_{0,\delta}\}_{1\leq \delta <\infty}\subset H^{s}(\mathcal{M})$ such that $u_{0,\delta}$ converges to $u_0$ in $H^{s}(\mathcal{M})$ as $\delta

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: Fractional Leibniz rule
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 24 more