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Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

Billel Guelmame, Didier Clamond, Stéphane Junca

Abstract

We prove in this note the local (in time) well-posedness of a broad class of $2 \times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond, Dutykh and Mitsotakis. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.

Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

Abstract

We prove in this note the local (in time) well-posedness of a broad class of symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond, Dutykh and Mitsotakis. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.
Paper Structure (3 sections, 7 theorems, 36 equations)

This paper contains 3 sections, 7 theorems, 36 equations.

Table of Contents

  1. Introduction and main results
  2. Local well-posedness of a general $2 \times 2$ system
  3. Proof of Theorems \ref{['existenceSV']} and \ref{['thm:blowup']}

Key Result

Theorem 1

Let $\tilde{m} \geqslant s \geqslant 2$, $0<g\in {C}^1([0,+\infty[)$, $d-\bar{d} \in {C}([0,+\infty], H^{s+1}) \cap {C}^1([0,+\infty], H^{s})$ and let $W_0=(\eta_0,u_0)^\top \in H^s$ satisfying $\inf_{x \in \mathds{R}} h_0(x) \geqslant h^* >0$, then there exist $T>0$ and a unique solution $W=(\eta,u

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Lemma 3