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A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$

Guorong Qu, Jianzhong Lu, Wei Deng

Abstract

This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$ and $1\leq r<\infty$, the solutions $\mathbf{S}_{t}^{\mathbfα}(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L^\infty_T(B^s_{2,r})$ norm to the unique solution of the Burgers equation as $α\rightarrow 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B^s_{2,\infty}(\R)$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L^\infty_T(B^s_{2,\infty})$ norm to the Burgers equation as $α\rightarrow 0$.

A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$

Abstract

This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data with and , the solutions of the Camassa-Holm equation exhibit convergence in the norm to the unique solution of the Burgers equation as . Contrary to this result, the present study demonstrates that for initial data the solutions of the Camassa-Holm equation fail to converge strongly in the norm to the Burgers equation as .

Paper Structure

This paper contains 3 sections, 10 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem

Key Result

Theorem 1.1

Assume that $s>\frac{3}{2}$ and $\alpha\in[0,1]$. Let $\mathbf{S}_{t}^{\mathbf{\alpha}}(v_0)$ be the solutions of alpha-c with the initial data $v_0\in B^s_{2,\infty}(\mathbb{R})$. Then there exist a initial data $u_0\in B^s_{2,\infty}(\mathbb{R})$ and a positive constant $\eta_0$ such that with $\alpha_n\rightarrow 0$ and $t_n\rightarrow 0$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 2.1: Littlewood-Paley decomposition, See B.C.D
  • Definition 2.1: See B.C.D
  • Definition 2.2: See B.C.D
  • Remark 2.1
  • Lemma 2.1: See B.C.D
  • Lemma 2.2: See B.C.D
  • Lemma 2.3: See B.C.D
  • Lemma 2.4: See C.L.L.W.L
  • Lemma 2.5: See C.L.L.W.L
  • ...and 6 more