Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scattering for the quintic generalized Benjamin-Bona-Mahony equation

Gong Chen, Yingmo Zhang

Abstract

We consider the quintic generalized Benjamin-Bona-Mahony equation $$ u_t-u_{xxt} + \partial_x\big(u + u^{5}\big)= 0,\qquad (t,x)\in \mathbb{R}_+ \times \mathbb{R}. $$ Using the space-time resonance method, we prove that sufficiently small and smooth solutions scatter to the linear flow. While the higher nonlinearity simplifies the treatment of nonresonant interactions compared to the quartic model in \cite{Morgan}, resonance analysis is more intricate. The resonance analysis occurs in a higher-dimensional geometric setting, and certain null or vanishing conditions present in the quartic case fail at specific resonance points. As a result, refined computations and precise estimates near the resonant set are required to close the bootstrap argument.

Scattering for the quintic generalized Benjamin-Bona-Mahony equation

Abstract

We consider the quintic generalized Benjamin-Bona-Mahony equation Using the space-time resonance method, we prove that sufficiently small and smooth solutions scatter to the linear flow. While the higher nonlinearity simplifies the treatment of nonresonant interactions compared to the quartic model in \cite{Morgan}, resonance analysis is more intricate. The resonance analysis occurs in a higher-dimensional geometric setting, and certain null or vanishing conditions present in the quartic case fail at specific resonance points. As a result, refined computations and precise estimates near the resonant set are required to close the bootstrap argument.
Paper Structure (21 sections, 45 theorems, 378 equations)

This paper contains 21 sections, 45 theorems, 378 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Acknowledgment
  5. Preliminaries
  6. Dispersive relations
  7. Littlewood--Paley projections
  8. Linear estimates
  9. Space--time resonance sets
  10. Determining resonances
  11. Proof of Proposition \ref{['prop:resonances']}
  12. Energy bounds and weighted estimates
  13. Energy estimates
  14. Weighted $L^2_x$ and $L^{\infty}_{\xi}$ bounds near resonances
  15. Anomalous Resonance
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix We do not claim the optimality for the regularity here. This is picked for the sake of convenience.$s \ge 100$. Consider the Cauchy problem eq:gBBM with $p = 4$ and initial condition There exists $\varepsilon_0 \in (0,1)$ such that if the initial data satisfy then the following hold

Theorems & Definitions (81)

  • Theorem 1.1
  • Definition 2.1: LP Projections
  • Lemma 2.2: LP Decomposition
  • Lemma 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 71 more