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Blowup analysis of a Camassa-Holm type equation with time-varying dissipation

Yonghui Zhou, Xiaowan Li, Shuguan Ji

Abstract

This paper is concerned with the local well-posedness, wave breaking, blow-up rate for a Camassa-Holm type equation with time-dependent weak dissipation. Firstly, we obtain the local well-posedness of solutions by using Kato's theory. Secondly, by using energy estimates, characteristic methods, and comparison principles, we derive two blowup criteria involving both pointwise gradient conditions and mixed amplitude-gradient conditions, and prove the blowup rate is universally $-2$. Our results extend wave breaking analysis to physically relevant variable dissipation regimes.

Blowup analysis of a Camassa-Holm type equation with time-varying dissipation

Abstract

This paper is concerned with the local well-posedness, wave breaking, blow-up rate for a Camassa-Holm type equation with time-dependent weak dissipation. Firstly, we obtain the local well-posedness of solutions by using Kato's theory. Secondly, by using energy estimates, characteristic methods, and comparison principles, we derive two blowup criteria involving both pointwise gradient conditions and mixed amplitude-gradient conditions, and prove the blowup rate is universally . Our results extend wave breaking analysis to physically relevant variable dissipation regimes.

Paper Structure

This paper contains 3 sections, 16 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Local well-posedness
  3. Wave breaking

Key Result

Theorem 2.1

Assume that (i)--(iii) hold. Given $z_{0}\in Y$, there exists a maximal $T>0$ depending only on $\|z_{0}\|_{Y}$ and unique solution $z$ to equation 201 such that Moreover, the map $z_{0}\mapsto z(\cdot,z_{0})$ is continuous from $Y$ to $C([0,T);Y)\cap C^{1}([0,T);X).$

Theorems & Definitions (26)

  • Theorem 2.1: Kato's theoremKato
  • Theorem 2.2: Local well-posedness
  • proof
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2: Kato1988
  • Lemma 3.3: Kato1988
  • Lemma 3.4: Constantin2002
  • Theorem 3.1
  • ...and 16 more