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The Cauchy problem and integrability of a modified Euler-Poisson equation

Feride Tiglay

TL;DR

The paper analyzes the periodic Cauchy problem for the modified Euler-Poisson equation (mEP) through three lenses: Sobolev-space well-posedness, analytic regularity, and bihamiltonian integrability. It proves local well-posedness in Sobolev spaces for $s>m/2+1$ (with an improved threshold to $s>3/2$ in 1D), via an ODE formulation on the diffeomorphism group and standard Sobolev estimates. It establishes a Cauchy–Kowalevski type result for analytic initial data, obtaining local analyticity in both space and time. In 1D, it reveals a bihamiltonian structure on a semidirect product, with two compatible Poisson brackets, confirming integrability and linking to Hunter–Zheng hierarchies.

Abstract

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in $H^{s} (T^{m})$ when $s>m/2+2$ and we improve the Sobolev index to $s>3/2$ for $m=1$. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space $ Diff \ltimes C^{\infty}(\tor)$ as a Hamiltonian equation, we concentrate to one space dimension ($m=1$) and show that the equation is bihamiltonian.

The Cauchy problem and integrability of a modified Euler-Poisson equation

TL;DR

The paper analyzes the periodic Cauchy problem for the modified Euler-Poisson equation (mEP) through three lenses: Sobolev-space well-posedness, analytic regularity, and bihamiltonian integrability. It proves local well-posedness in Sobolev spaces for (with an improved threshold to in 1D), via an ODE formulation on the diffeomorphism group and standard Sobolev estimates. It establishes a Cauchy–Kowalevski type result for analytic initial data, obtaining local analyticity in both space and time. In 1D, it reveals a bihamiltonian structure on a semidirect product, with two compatible Poisson brackets, confirming integrability and linking to Hunter–Zheng hierarchies.

Abstract

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in when and we improve the Sobolev index to for . We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space as a Hamiltonian equation, we concentrate to one space dimension () and show that the equation is bihamiltonian.

Paper Structure

This paper contains 3 sections, 9 theorems, 123 equations.

Table of Contents

  1. Local well-posedness in Sobolev spaces
  2. Analytic regularity
  3. Bihamiltonian structure and integrability

Key Result

Theorem 1

For s>m/2+1, given any initial data (n_{0}, v_{0}) \in H^{s-1}(\mathbb{T}^{m},\mathbb{R}) \times H^{s}(\mathbb{T}^{m},\mathbb{R}^{m}), there exists a T>0 and a unique solution (n,v) to the Cauchy problem for the modified Euler-Poisson equation (mEP) such that and and the solution (n,v) depends continuously on the initial data (n_{0},v_{0}).

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 4