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Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence

Jinlu Li, Yanghai Yu

Abstract

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the Hölder spaces from $C^{k,α}$ to $L_T^\infty C^{k,α}$ with $k\in \mathbb{N}^+$ and $α\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,α}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.

Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence

Abstract

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from to for with and in the Hölder spaces from to with and , which later covers particularly the ill-posedness of solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.

Paper Structure

This paper contains 17 sections, 20 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Motivations and Previous Results
  3. Main results
  4. Difficulties and Strategies to the proof of Theorem \ref{['th1']}
  5. Organization of our paper
  6. Preliminaries
  7. Non-Continuous Dependence of 1-D Transport Equation
  8. Construction of Initial Data
  9. Technical Lemmas
  10. Constructions of Approximation Solutions
  11. Estimations of Approximate Solutions
  12. Non-Continuous Dependence
  13. Proof of Theorem \ref{['th1']}: Periodic case
  14. Proof of Theorem \ref{['th1']}: Non-periodic case
  15. Constructions of Approximation Solutions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $1 \leq p, r \leq \infty$ and $s \in \mathbb{R}$ be such that $B_{p, r}^s(\mathbb{R}^d) \hookrightarrow C^{0,1}(\mathbb{R}^d)(i.e.\ s>d/ p+1\ \text{or}\ s=d/ p+1,r=1)$. There exists a constant $c$, depending only on $s, p, r$, and $d$, such that for all divergence-free data $u_0 \in B_{p, r}^s(\ Namely, if $r<\infty$ (resp., $r=\infty$ ), then $u$ is continuous (resp., weakly continuous) in ti

Theorems & Definitions (41)

  • Theorem 1.1: Theorem 7.1, BCD
  • Theorem 1.2
  • Corollary 1.1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.2
  • Definition 2.1: BCD
  • Remark 2.1
  • Lemma 2.1: BCD
  • Lemma 2.2: guo
  • ...and 31 more