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On two systems of Burgers type arising in nonlinear wave interaction

Diego Alonso-Orán, Rafael Granero-Belinchón

Abstract

In this note we study the well-posedness of two systems of Burgers type arising in nonlinear wave interactions. The first model describes the interaction of a Burger's bore with the classical Korteweg-de Vries equation while the second exemplify the interaction of weak sound waves and entropy waves with small amplitudes. For the former, we show the local existence and uniqueness of solutions in Sobolev spaces and Wiener-type spaces. For the latter, we provide an elementary proof of finite time singularity.

On two systems of Burgers type arising in nonlinear wave interaction

Abstract

In this note we study the well-posedness of two systems of Burgers type arising in nonlinear wave interactions. The first model describes the interaction of a Burger's bore with the classical Korteweg-de Vries equation while the second exemplify the interaction of weak sound waves and entropy waves with small amplitudes. For the former, we show the local existence and uniqueness of solutions in Sobolev spaces and Wiener-type spaces. For the latter, we provide an elementary proof of finite time singularity.

Paper Structure

This paper contains 8 sections, 3 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Notations and functional spaces
  3. Main results
  4. Outline
  5. Proof of Theorem \ref{['teobkdv1']} and Theorem \ref{['teobkdv2']}
  6. Proof of Theorem \ref{['teobkdv1']}
  7. Proof of Theorem \ref{['teobkdv2']}
  8. Proof of the Theorem \ref{['teomrs']}

Key Result

Theorem 1.1

Let $\nu,\gamma>0$. For $(u_0,v_0)\in H^3\times H^4$ with zero mean initial data, there exists a time $0<T_{max}$ and a unique solution of the initial value problem of diss:BKdV with $u(x,0)=u_{0}(x), v(x,0)=v_{0}(x), \ x\in\mathbb{T}$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7