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Refined wave breaking for the generalized Fornberg-Whitham equation

Jean-Claude Saut, Yuexun Wang

Abstract

This paper considers a class of non-local equations that are weakly dispersive perturbations of the inviscid Burgers equation, which includes the Fornberg-Whitham equation as a special case. We precise the known results on finite time blow-up (shock formation) by constructing a blowup solution which displays a `shock-like' singularity (called wave breaking) at one single point. Moreover, this solution converges asymptotically in the self-similar variables to a stable self-similar solution of the inviscid Burgers equation, and also possesses a Hölder $C^{1/3}$ regularity at the blowup point.

Refined wave breaking for the generalized Fornberg-Whitham equation

Abstract

This paper considers a class of non-local equations that are weakly dispersive perturbations of the inviscid Burgers equation, which includes the Fornberg-Whitham equation as a special case. We precise the known results on finite time blow-up (shock formation) by constructing a blowup solution which displays a `shock-like' singularity (called wave breaking) at one single point. Moreover, this solution converges asymptotically in the self-similar variables to a stable self-similar solution of the inviscid Burgers equation, and also possesses a Hölder regularity at the blowup point.
Paper Structure (36 sections, 14 theorems, 257 equations)

This paper contains 36 sections, 14 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main results
  3. Notation and conventions.
  4. The self-similar transformation
  5. The stable blowup self-similar Burgers profile
  6. Equations on $\partial_X^n\widetilde{U}$
  7. Constraints on $U$ and equations of $\xi, \tau$ and $\kappa$
  8. Assumptions on the initial data
  9. Main results
  10. Bootstrap assumptions
  11. Bootstrap assumptions for the self-similar variables
  12. Bootstrap assumptions for the dynamic modulations variables
  13. Consequences of the bootstrap assumptions
  14. $L^{2}$ estimates
  15. Low-order $L^{2}$ bounds
  16. ...and 21 more sections

Key Result

Theorem 2.1

Let $\alpha<0$. There exist a sufficiently large $M>0$ and a sufficiently small $\varepsilon=\varepsilon(M,\alpha)>0$ such that if the initial data $u_{0}$ satisfies eq:3.7-eq:4.3, then there exists a unique solution $u\in C([-\varepsilon, T_*), H^5(\mathbb{R}))$ to the Cauchy problem of eq:1.1 such respectively. (II) $u$ is bounded: $\|u(\cdot, t)\|_{L^\infty}\leq M$ for all $t\in [-\varepsilon,

Theorems & Definitions (25)

  • Theorem 2.1
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 15 more