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Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations

Yiyao Lian, Zhenyu Wan, Zhaoyang Yin

Abstract

In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations $u_t-u_{xxt}+(f\left(u\right))_x-(f\left(u\right))_{xxx}+\left(g\left(u\right)+\frac{f^{\prime\prime}\left(u\right)}{2}u_x^2\right)_x+λ\left(u-u_{xx}\right)=0$. This includes the weakly dissipative Camassa-Holm equation and the weakly dissipative hyperelastic rod wave equation as special cases. Specifically, we establish three global existence results: one concerning the energy conservative weak solutions in a time-weighted $H^1$ space, and the other two concerning strong solutions, which include the cases of small initial data and sign-changing initial data. Our results recover and extend many known results for several classical models.

Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations

Abstract

In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations . This includes the weakly dissipative Camassa-Holm equation and the weakly dissipative hyperelastic rod wave equation as special cases. Specifically, we establish three global existence results: one concerning the energy conservative weak solutions in a time-weighted space, and the other two concerning strong solutions, which include the cases of small initial data and sign-changing initial data. Our results recover and extend many known results for several classical models.
Paper Structure (11 sections, 12 theorems, 141 equations)

This paper contains 11 sections, 12 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Main results and strategy
  3. Organization and notation
  4. Preliminaries
  5. Global conservative weak solutions
  6. An equivalent semilinear system
  7. Global solutions of the semilinear system
  8. Global conservative weak solutions for the original equation
  9. Global strong solutions
  10. Global strong solutions for small data
  11. Global strong solutions for sign-changing data

Key Result

Theorem 1.2

Let $\bar{u}\in H^1(\mathbb{R})$. Then the Cauchy problem eq;u1 has a global conservative weak solution in the sense of Definition def2.1. Furthermore, if there is a sequence of the initial value $\bar{u}_{n}$ satisfying $\| \bar{u}_{n}-\bar{u}\|_{H^{1}}\to 0$ as $n\to \infty$, then the correspondin

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 10 more