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Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited

Jinlu Li, Yanghai Yu, Weipeng Zhu

Abstract

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\frac{1}{2}$ and $1\leq p, r< \infty$, which improves the previous works [Himonas et al., Asian J. Math., 11 (2007)], [Li et al., J. Differ. Equ., 269 (2020)] and [Li et al., J. Math. Fluid Mech., 23 (2021)]. Furthermore, we present a strengthening of our previous work in [Li et al., J. Differ. Equ., 269 (2020)] and prove that the data-to-solution map for the Camassa--Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R})$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa--Holm and Degasperis--Procesi equations.

Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited

Abstract

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on the initial data in Besov spaces with and , which improves the previous works [Himonas et al., Asian J. Math., 11 (2007)], [Li et al., J. Differ. Equ., 269 (2020)] and [Li et al., J. Math. Fluid Mech., 23 (2021)]. Furthermore, we present a strengthening of our previous work in [Li et al., J. Differ. Equ., 269 (2020)] and prove that the data-to-solution map for the Camassa--Holm equation is nowhere uniformly continuous in with and . The method applies also to the b-family of equations which contain the Camassa--Holm and Degasperis--Procesi equations.
Paper Structure (7 sections, 20 theorems, 104 equations)

This paper contains 7 sections, 20 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Littlewood-Paley analysis
  5. Proof of Theorem \ref{['th-camassa']}
  6. Technical Lemmas
  7. Proof of Theorem \ref{['th4']}

Key Result

Theorem 1.1

Denote $U_R\equiv\{u_0\in B_{p,r}^s: \|u_0\|_{B^{s}_{p,r}}\leq R\}$ for any $R>0$. Assume that $(s,p,r)$ satisfies Then the data-to-solution map of the Cauchy problem CH--CH1 is not uniformly continuous from any bounded subset $U_R$ in $B^s_{p,r}$ into $\mathcal{C}([0,T];B^s_{p,r})$. More precisely, there exists two sequences of solutions $\mathbf{S}_t(f_n+g_n)$ and $\mathbf{S}_t(f_n)$ such that

Theorems & Definitions (39)

  • Theorem 1.1: Lyz
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.3: Nowhere uniformly continuous
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 29 more