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The failure of Hölder regularity of solutions for the Camassa--Holm type equation in Besov spaces

Jinlu Li, Yanghai Yu, Weipeng Zhu

Abstract

It is proved that if $u_0\in B^s_{p,r}$ with $s>1+\frac1p, (p,r)\in[1,+\infty]\times[1,+\infty)$ or $s=1+\frac1p, \ (p,r)\in[1,+\infty)\times \{1\}$, the solution of the Camassa--Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$. In the paper, we show that the continuity of the solution can not be improved to the Hölder continuity. Precisely speaking, the solution of the Camassa--Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^α([0,T];B^s_{p,r})$ with any $α\in(0,1)$.

The failure of Hölder regularity of solutions for the Camassa--Holm type equation in Besov spaces

Abstract

It is proved that if with or , the solution of the Camassa--Holm equation belongs to . In the paper, we show that the continuity of the solution can not be improved to the Hölder continuity. Precisely speaking, the solution of the Camassa--Holm equation belongs to but not to with any .
Paper Structure (7 sections, 10 theorems, 60 equations, 1 table)

This paper contains 7 sections, 10 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main Theorems
  4. Construction of Initial Data
  5. Error Estimates
  6. Proof of Theorem \ref{['th1']}
  7. Proof of Theorem \ref{['th2']}

Key Result

Theorem 1.1

Assume that $(s,p,r)$ satisfies that For any $\alpha\in(0,1)$, there exits $u_0\in B^s_{p,r}(\mathbb{R})$ such that the data-to-solution map $u_0\mapsto \mathbf{S}_{t}(u_0)\in \mathcal{C}([0,T];B^s_{p,r})$ of the Cauchy problem CH satisfies

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: see BCD
  • Definition 2.2: see BCD
  • Lemma 2.1: see BCD
  • Lemma 2.2: see BCD
  • Lemma 2.3: see BCD
  • ...and 6 more