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The local well-posedness, blow-up phenomena and ill-posedness of a new fifth-order Camassa-Holm type equation

Yiyao Lian, Zhaoyang Yin

TL;DR

This work analyzes a new fifth-order Camassa–Holm type equation with cubic nonlinearities and higher-order dispersion. It develops local well-posedness in Besov spaces via a transport-based construction and Moser-type estimates, establishes a sharp blow-up criterion, and derives a precise blow-up time using conservation laws and a Lagrangian framework. It further demonstrates ill-posedness at the critical level $H^{\frac{1}{2}}$ through norm inflation, highlighting instability of the dynamics at minimal regularity. Together, these results extend the CH family to higher-order dispersion, providing rigorous conditions for global behavior, finite-time blow-up, and ill-posedness at critical regularity.

Abstract

In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \cite{Li.Z}. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation in Besov spaces. Secondly, we obtain blow-up criteria. Building upon this, by utilizing the conservation laws and establishing local boundedness, we derive a blow-up result that precisely determines the blow-up time. Finally, the ill-posedness of the new fifth-order Camassa-Holm type equation in the critical Sobolev space $H^{\frac{1}{2}}$ is established via a norm inflation argument.

The local well-posedness, blow-up phenomena and ill-posedness of a new fifth-order Camassa-Holm type equation

TL;DR

This work analyzes a new fifth-order Camassa–Holm type equation with cubic nonlinearities and higher-order dispersion. It develops local well-posedness in Besov spaces via a transport-based construction and Moser-type estimates, establishes a sharp blow-up criterion, and derives a precise blow-up time using conservation laws and a Lagrangian framework. It further demonstrates ill-posedness at the critical level through norm inflation, highlighting instability of the dynamics at minimal regularity. Together, these results extend the CH family to higher-order dispersion, providing rigorous conditions for global behavior, finite-time blow-up, and ill-posedness at critical regularity.

Abstract

In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \cite{Li.Z}. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation in Besov spaces. Secondly, we obtain blow-up criteria. Building upon this, by utilizing the conservation laws and establishing local boundedness, we derive a blow-up result that precisely determines the blow-up time. Finally, the ill-posedness of the new fifth-order Camassa-Holm type equation in the critical Sobolev space is established via a norm inflation argument.
Paper Structure (5 sections, 16 theorems, 132 equations)

This paper contains 5 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local well-posedness
  4. Blow up
  5. Ill-posedness

Key Result

Proposition 2.1

(The Littlewood-Paley decomposition) BCD Let $\mathcal{C}$ be the annulus $\{\xi\in\mathbb{R}^d:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$. There exist radial functions $\chi$ and $\varphi$, valued in the interval $[0,1]$, belonging respecitvely to $\mathcal{D}(B(0,\frac{4}{3}))$ and $\mathcal{D}(\mathc

Theorems & Definitions (23)

  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9: A priori estimates in Besov spaces
  • Lemma 2.10
  • ...and 13 more