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Hamiltonian regularisation of the unidimensional barotropic Euler equations

Billel Guelmame, Didier Clamond, Stéphane Junca

Abstract

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh. This system is Galilean invariant, linearly non-dispersive and conserves formally an $H^1$-like energy. In this paper, we generalise this regularisation for the barotropic Euler system preserving the same properties. We prove the local (in time) well-posedness of the regularised barotropic Euler system and a periodic generalised two-component Hunterr-Saxton system. We also show for both systems that if singularities appear in finite time, they are necessary in the first derivatives.

Hamiltonian regularisation of the unidimensional barotropic Euler equations

Abstract

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh. This system is Galilean invariant, linearly non-dispersive and conserves formally an -like energy. In this paper, we generalise this regularisation for the barotropic Euler system preserving the same properties. We prove the local (in time) well-posedness of the regularised barotropic Euler system and a periodic generalised two-component Hunterr-Saxton system. We also show for both systems that if singularities appear in finite time, they are necessary in the first derivatives.
Paper Structure (23 sections, 6 theorems, 147 equations)

This paper contains 23 sections, 6 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Equations for barotropic perfect fluids
  3. Cauchy--Lagrange equation
  4. Conservation laws
  5. Jump conditions
  6. Variational formulations
  7. Regularised barotropic flows
  8. Modified Lagrangian
  9. Linearised equations
  10. Equations of motion
  11. Secondary equations
  12. Rankine--Hugoniot conditions
  13. Hamiltonian formulation
  14. Steady motions
  15. Local analysis of steady solution
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\tilde{m} \geqslant s \geqslant 2$, $\tilde{m}$ be an integer, $P, \mathscr{A} \in {C}^{\tilde{m}+4}(]0,+\infty[)$ such that $P'(\rho)~>~0$, $\mathscr{A}'(\rho)>0$ for $\rho>0$. Let also $W_0=(\tilde{\rho}_0,u_0)^\top \in H^s$ satisfying $\inf_{x \in \mathds{R}} \rho_0(x) >~\rho^{*}$, then ther Moreover, if the maximal existence time $T_{max}< +\infty$, then

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • ...and 1 more