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Ill-posedness and global solution for the $b$-equation

Yingying Guo, Weikui Ye

Abstract

In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of the $b$-equation is established when $b\geq1$ or $b\leq1.$ It's worth noting that our global result is a new result which doesn't need the condition that $m_{0}(x)$ keeps its sign. For $s<\frac32,$ it is shown (see [13]) that the Cauchy problem of the $b$-equation is ill-posed in Sobolev space $H^{s}(\mathbb{R})$ when $b>1$ or $b<1.$ In the present paper, for $s=\frac32,$ we prove that the Cauchy problem of the $b$-equation is also ill-posed in $H^{\frac32}(\mathbb{R})$ in the sense of norm inflation by constructing a class of special initial data when $b\neq1.$

Ill-posedness and global solution for the $b$-equation

Abstract

In this paper, we consider the Cauchy problem for the -equation. Firstly, for if and the global solutions of the -equation is established when or It's worth noting that our global result is a new result which doesn't need the condition that keeps its sign. For it is shown (see [13]) that the Cauchy problem of the -equation is ill-posed in Sobolev space when or In the present paper, for we prove that the Cauchy problem of the -equation is also ill-posed in in the sense of norm inflation by constructing a class of special initial data when
Paper Structure (4 sections, 6 theorems, 39 equations, 1 table)

This paper contains 4 sections, 6 theorems, 39 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['global']}
  4. The proof of Theorem \ref{['ill']}

Key Result

Theorem 1.1

Suppose $u_{0}(x)\in H^{s}(\mathbb{R}),\ s>\frac{3}{2}$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}).$ For $b\geq 1,$ if there is a $x_{0}\in\mathbb{R}$ such that or for $b\leq1,$ if there exists a $x_{0}\in\mathbb{R}$ such that then the correspending solution to bb exists globally.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 2.1: See book
  • Lemma 3.1: See ey1glt
  • Proposition 3.1: See ey1
  • Lemma 3.2
  • proof
  • proof : The proof of Theorem \ref{['global']}:
  • ...and 1 more