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Forced oscillation of a damped BBM equation posed on whole line in low regularity spaces

Chun Ho Lau, Taige Wang

TL;DR

This work studies forced damped BBM dynamics on $\mathbb{R}$ in low-regularity spaces $H^\ell$, $\ell\in[0,1)$. It employs the $I$-method with the operator $I_N$ to lift to an $H^1$-based framework and proves local and global well-posedness, along with existence and stability of time-periodic solutions under time-periodic forcing. Key contributions include bilinear/trilinear estimates and a fixed-point scheme that establish mild solutions in $Y^\ell_{\tau,T}$, the construction of time-periodic states, and exponential stability results that persist globally under small forcing. The results extend damped BBM/KdV-type dynamics to rough data, providing a rigorous handle on long-time behavior and asymptotic periodicity in low-regularity settings.

Abstract

In this manuscript, we would established in low regularity spaces $H^\ell, \ell\in [0,1)$, the existence and stability results of time-periodic solution of 1D Cauchy problem of forced damped Benjamin-Bona-Mahony equation (BBM). We use estimates from I-energy method to derive needed estimates in $H^\ell$ for the linearized problem, then convection term will be treated as perturbation of linear problem such that original Cauchy problem is solved.

Forced oscillation of a damped BBM equation posed on whole line in low regularity spaces

TL;DR

This work studies forced damped BBM dynamics on in low-regularity spaces , . It employs the -method with the operator to lift to an -based framework and proves local and global well-posedness, along with existence and stability of time-periodic solutions under time-periodic forcing. Key contributions include bilinear/trilinear estimates and a fixed-point scheme that establish mild solutions in , the construction of time-periodic states, and exponential stability results that persist globally under small forcing. The results extend damped BBM/KdV-type dynamics to rough data, providing a rigorous handle on long-time behavior and asymptotic periodicity in low-regularity settings.

Abstract

In this manuscript, we would established in low regularity spaces , the existence and stability results of time-periodic solution of 1D Cauchy problem of forced damped Benjamin-Bona-Mahony equation (BBM). We use estimates from I-energy method to derive needed estimates in for the linearized problem, then convection term will be treated as perturbation of linear problem such that original Cauchy problem is solved.
Paper Structure (4 sections, 9 theorems, 46 equations)

This paper contains 4 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Proof of main theorems

Key Result

Theorem 2.3

Given $\tau\ge0, T>0, \phi\in H^\ell, f\in L^2$ for $\ell\in[0, 1)$, if there exists a constant $\delta>0$ and holds smallness condition $\|\phi\|_{H^{\ell}}+ \sup_{t\ge 0}\|f(t)\|\le \delta$, there holds local well-posedness of solution $u$ to Cauchy problem (1-1) where $u$ satisfies

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 4 more