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On the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation

Younghun Hong, Junyeong Jang, Changhun Yang

TL;DR

The paper analyzes the Benjamin-Bona-Mahony (BBM) regularization of the Korteweg-de Vries (KdV) equation for long waves, proving convergence of BBM$_\epsilon$ to KdV for energy-class solutions and extending the time interval of validity via conserved quantities. The authors develop a Bourgain-space (Fourier restriction) framework tailored to the rescaled BBM flow, establish uniform bilinear estimates, and recast the BBM–KdV comparison in integral form to control the difference. They prove local-in-time convergence for $1\le s\le 5$ with an error $\epsilon^{2s/5}$ and, using conservation laws, obtain global-in-time convergence for $s=1$ with an exponential-in-time bound, yielding a logarithmic extension of the validity interval as $\epsilon\to 0$. Overall, the results validate the BBM regularization as a robust, energy-class approximation to KdV and clarify how conserved quantities extend the applicability of the BBM model in long-wave regimes.

Abstract

The Benjamin-Bona-Mahony equation (BBM) is introduced as a regularization of the Korteweg-de Vries equation (KdV) for long water waves \cite{BBM1972}. In this paper, we establish the convergence from the BBM to the KdV for energy class solutions. As a consequence, employing the conservation laws, we extend the known temporal interval of validity for the BBM regularization.

On the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation

TL;DR

The paper analyzes the Benjamin-Bona-Mahony (BBM) regularization of the Korteweg-de Vries (KdV) equation for long waves, proving convergence of BBM to KdV for energy-class solutions and extending the time interval of validity via conserved quantities. The authors develop a Bourgain-space (Fourier restriction) framework tailored to the rescaled BBM flow, establish uniform bilinear estimates, and recast the BBM–KdV comparison in integral form to control the difference. They prove local-in-time convergence for with an error and, using conservation laws, obtain global-in-time convergence for with an exponential-in-time bound, yielding a logarithmic extension of the validity interval as . Overall, the results validate the BBM regularization as a robust, energy-class approximation to KdV and clarify how conserved quantities extend the applicability of the BBM model in long-wave regimes.

Abstract

The Benjamin-Bona-Mahony equation (BBM) is introduced as a regularization of the Korteweg-de Vries equation (KdV) for long water waves \cite{BBM1972}. In this paper, we establish the convergence from the BBM to the KdV for energy class solutions. As a consequence, employing the conservation laws, we extend the known temporal interval of validity for the BBM regularization.
Paper Structure (20 sections, 13 theorems, 106 equations)

This paper contains 20 sections, 13 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Ideas of the proof
  5. Organization of the paper
  6. Notations
  7. Acknowledgement
  8. Preliminaries
  9. Bilinear estimates for the rescaled linear BBM flow
  10. Bilinear estimate for the rescaled linear BBM flow
  11. Reduction to the integral estimate
  12. Proof of Proposition \ref{['Lem:bilinear for BBM']}
  13. Bilinear estimate for the difference in the nonlinear terms
  14. Uniform bounds for nonlinear solutions
  15. Proof of the main results (Theorem \ref{['thm: main result 1.1']} and \ref{['thm: main result 1.2']})
  16. ...and 5 more sections

Key Result

Theorem 1.2

Suppose that $1\leq s\leq 5$ and and let $u_\epsilon(t)\in C_t(\mathbb{R};H_x^s)$ (resp., $w(t)\in C_t(\mathbb{R};H_x^s)$) be the unique solution to the $\textup{BBM}_\epsilon$eq: BBM (resp., the KdV eq: KdV) with the initial data $u_{\epsilon, 0}$ (resp., $w_0$). Then, there exists $T=T(R)>0$, independent of $\epsilon\in(0,1]$, su

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 1.2: Local-in-time BBM regularization of the KdV
  • Remark 1.3
  • Theorem 1.4: Global-in-time BBM regularization for the KdV
  • Remark 1.5
  • Remark 1.6: Reformulation of Theorem \ref{['thm: main result 1.1']} and \ref{['thm: main result 1.2']}
  • Lemma 2.1: Basic properties of the Fourier restriction norm
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1: Bilinear estimate for the rescaled linear BBM flow
  • ...and 18 more