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The failure of Hölder regularity of solutions for the Euler-Poincaré equations in Besov spaces

Guorong Qu, Min Li

Abstract

In this paper, we investigate the continuity of solution to the Euler-Poincaré equations. We show that the continuity of the solution cannot be improved to the Hölder continuity. That is, the solution of the Euler-Poincaré equations with initial data $u_0\in B^s_{p,r}$ belongs to $\mathcal{C}([0,T];B^s_{p,r}(\mathbb R^d))$ but not to $\mathcal{C}^α([0,T];B^s_{p,r}(\mathbb R^d))$ with any $α\in(0,1)$.

The failure of Hölder regularity of solutions for the Euler-Poincaré equations in Besov spaces

Abstract

In this paper, we investigate the continuity of solution to the Euler-Poincaré equations. We show that the continuity of the solution cannot be improved to the Hölder continuity. That is, the solution of the Euler-Poincaré equations with initial data belongs to but not to with any .
Paper Structure (3 sections, 5 theorems, 44 equations)

This paper contains 3 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem

Key Result

Theorem 1.1

Let $d\geq 2$. Assume that $(s,p,r)$ satisfies For any $\alpha\in(0,1)$, there exits $u_0\in B^s_{p,r}(\mathbb{R}^d)$ such that the data-to-solution map $u_0\mapsto \mathbf{S}_{t}(u_0)\in \mathcal{C}([0,T];B^s_{p,r})$ of the Cauchy problem 0 satisfies

Theorems & Definitions (6)

  • Theorem 1.1
  • Definition 2.1: bcd
  • Lemma 2.1: bcd and Lemma 2.7, Li-Yin17
  • Lemma 2.2: see bcd
  • Lemma 3.1
  • Proposition 3.1