Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Refined wave breaking for the one-dimensional nonlinear shallow water equations

Pingchun Liu, Jean-Claude Saut, Shihan Sun, Yuexun Wang

Abstract

This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography parameters, under a non-cavitation condition which excludes the scenario that the solutions have compact support. We construct smooth initial data with finite $\dot{H}^5$-norm such that the $L^\infty$-norm of the spatial derivative of the solution blows up at one single point in finite time with a precise blowup profile.

Refined wave breaking for the one-dimensional nonlinear shallow water equations

Abstract

This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography parameters, under a non-cavitation condition which excludes the scenario that the solutions have compact support. We construct smooth initial data with finite -norm such that the -norm of the spatial derivative of the solution blows up at one single point in finite time with a precise blowup profile.
Paper Structure (40 sections, 5 theorems, 224 equations)

This paper contains 40 sections, 5 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. The Reformulation of the Main Problem
  3. Riemann invariants and self-similar transforms
  4. The stable blowup self-similar Burgers profile
  5. Evolution of equations of higher order derivatives
  6. Higher-order derivatives for the $(W, Z)$-system
  7. Higher-order derivatives for evolution of $\widetilde{W}$
  8. Constraints on $W$ and equations of the dynamic modulation variables
  9. Main results
  10. Initial data on self-similar variables
  11. Main results
  12. Sharp lifespan at times of order $\mathcal{O}(1/\max\{\varepsilon,\beta^* \})$
  13. Bootstrap assumptions
  14. Bootstrap assumptions for the self-similar variables.
  15. Bootstrap assumptions for the dynamic modulations variables.
  16. ...and 25 more sections

Key Result

Theorem 3.1

There exist a sufficiently large $M>0$ and a sufficiently small $\delta=\delta(M)>0$ such that if the initial data $(w_{0},z_{0}) \in \dot{H}^{5}(\mathbb{R})$ satisfies eq:3.1-eq:3.8 and $b(x)\in\dot{H}^{6}(\mathbb{R})$, then the unique solution to the Cauchy problem of eq:2.2 with $(w_{0},z_{0})$ forms a wave breaking in finite time. Furthermore,

Theorems & Definitions (7)

  • Theorem 3.1
  • Corollary 1
  • Lemma 1: MR1477408
  • Lemma 2
  • proof
  • Lemma 3
  • proof