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Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $λ$-family equations

Geng Chen, Yannan Shen, Shihui Zhu

Abstract

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as $λ$-family equations, where $λ$ is the power of nonlinear wave speed. The $λ$-family equations include Camassa-Holm equation ($λ=1$) and Novikov equation ($λ=2$) modelling water waves, where solutions generically form finite time cusp singularities, or in another word, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent $1- \frac{1}{2λ}$. The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $λ$-family equations

Abstract

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as -family equations, where is the power of nonlinear wave speed. The -family equations include Camassa-Holm equation () and Novikov equation () modelling water waves, where solutions generically form finite time cusp singularities, or in another word, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent . The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

Paper Structure

This paper contains 17 sections, 5 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Notations, basic setups and energy
  3. A general class of equations
  4. Key idea
  5. Main result
  6. Energy and uniform bounds on $P$ $P_x$, $Q$ and $Q_x$
  7. The semi-linear system
  8. Local existence of semi-linear system \ref{['S-1']}-\ref{['S-2']}
  9. Global existence of semi-linear system \ref{['S-1']}-\ref{['S-2']}
  10. 1.
  11. 2.
  12. 3.
  13. 4.
  14. 5.
  15. Global existence of the original equation
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\lambda=1$ or $\lambda\geq 2$. Suppose that $u_0\in H^1(\mathbb{R})\cap W^{1,2\lambda}(\mathbb{R})$ is an absolute continuous function on $x$. Then the initial value problem (E) (Ei) admits a weak solution $u(t,x)$, in the sense of Definition def1, defined for all $(t,x)\in\mathbb{R}^+\times\ma

Theorems & Definitions (12)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • Proposition 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 2 more