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Well-posedness and no-uniform dependence for the Euler-Poincaré equations in Triebel-Lizorkin spaces

Yuanhua Zhong, Jianzhong Lu, Min Li, Jinlu Li

Abstract

In this paper, we study the Cauchy problem of the Euler-Poincaré equations in $\R^d$ with initial data belonging to the Triebel-Lizorkin spaces. We prove the local-in-time unique existence of solutions to the Euler-Poincaré equations in $F^s_{p,r}(\R^d)$. Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.

Well-posedness and no-uniform dependence for the Euler-Poincaré equations in Triebel-Lizorkin spaces

Abstract

In this paper, we study the Cauchy problem of the Euler-Poincaré equations in with initial data belonging to the Triebel-Lizorkin spaces. We prove the local-in-time unique existence of solutions to the Euler-Poincaré equations in . Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.
Paper Structure (3 sections, 10 theorems, 81 equations)

This paper contains 3 sections, 10 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local well-posedness and no-uniform dependence

Key Result

Theorem 1.1

Let $d\geq 1$ and $1< p, r<\infty$. Assume that $u_0\in F^{s}_{p,r}(\mathbb{R}^d)$ with $s>\max\{\frac{3}{2},1+\frac{d}{p}\}$. Then there exists some time $T>0$ such that

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Proposition 2.1: bcd
  • Definition 2.1: bcd
  • Definition 2.2: Triebel
  • Remark 2.1
  • Lemma 2.1: bcdTriebel
  • Lemma 2.2: bcdTriebel
  • Lemma 2.3: bcdchae
  • ...and 7 more