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On the continuous properties for the 3D incompressible rotating Euler equations

Jinlu Li, Yanghai Yu, Neng Zhu

TL;DR

This work analyzes the Cauchy problem for the 3D incompressible Euler equations with the Coriolis force in $\mathbb{R}^3$, proving local existence and uniqueness in Besov spaces $B^s_{p,r}$ under standard regularity, while revealing three ill-posedness phenomena: non-uniform dependence on initial data, failure of Hölder continuity in time, and discontinuity of the solution map at $t=0$ in $B^s_{p,\infty}$. The authors develop a trio of technical mechanisms—a perturbation-based construction for non-uniform dependence, a high-frequency oscillatory bootstrap with commutator estimates to destroy time Hölder continuity, and a tailored initial-data scheme to force zero-time discontinuity—unifying these results within the Besov framework and extending prior HM1 insights to general Besov spaces. The findings highlight that even with rotation-induced dispersion, the flow map can exhibit strong forms of ill-posedness, guiding expectations about well-posedness in borderline Besov scales and suggesting refined thresholds for $p$-based Besov spaces. Overall, the paper advances the understanding of continuous properties of rotating Euler dynamics and clarifies the delicate balance between nonlinearity and dispersion in high-regularity function spaces.

Abstract

In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in $B^s_{p,r}(\R^3)$. Then we prove that the Cauchy problem is ill-posed in two different sense: (1) the solution of this system is not uniformly continuous dependence on the initial data in the same Besov spaces, which extends the recent work of Himonas-Misiołek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the general Besov spaces framework; (2) the solution of this system cannot be Hölder continuous in time variable in the same Besov spaces. In particular, the solution of the system is discontinuous in the weaker Besov spaces at time zero. To the best of our knowledge, our work is the first one addressing the issue on the failure of Hölder continuous in time of solution to the classical Euler equations with(out) the Coriolis force.

On the continuous properties for the 3D incompressible rotating Euler equations

TL;DR

This work analyzes the Cauchy problem for the 3D incompressible Euler equations with the Coriolis force in , proving local existence and uniqueness in Besov spaces under standard regularity, while revealing three ill-posedness phenomena: non-uniform dependence on initial data, failure of Hölder continuity in time, and discontinuity of the solution map at in . The authors develop a trio of technical mechanisms—a perturbation-based construction for non-uniform dependence, a high-frequency oscillatory bootstrap with commutator estimates to destroy time Hölder continuity, and a tailored initial-data scheme to force zero-time discontinuity—unifying these results within the Besov framework and extending prior HM1 insights to general Besov spaces. The findings highlight that even with rotation-induced dispersion, the flow map can exhibit strong forms of ill-posedness, guiding expectations about well-posedness in borderline Besov scales and suggesting refined thresholds for -based Besov spaces. Overall, the paper advances the understanding of continuous properties of rotating Euler dynamics and clarifies the delicate balance between nonlinearity and dispersion in high-regularity function spaces.

Abstract

In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in . Then we prove that the Cauchy problem is ill-posed in two different sense: (1) the solution of this system is not uniformly continuous dependence on the initial data in the same Besov spaces, which extends the recent work of Himonas-Misiołek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the general Besov spaces framework; (2) the solution of this system cannot be Hölder continuous in time variable in the same Besov spaces. In particular, the solution of the system is discontinuous in the weaker Besov spaces at time zero. To the best of our knowledge, our work is the first one addressing the issue on the failure of Hölder continuous in time of solution to the classical Euler equations with(out) the Coriolis force.

Paper Structure

This paper contains 11 sections, 12 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Known Well/Ill-posedness results
  3. Motivation
  4. Main Result
  5. Strategies to the proof
  6. Organization of our paper
  7. Preliminaries
  8. Proof of Theorem \ref{['th2']}
  9. Proof of Theorem \ref{['th3']}
  10. Proof of Theorem \ref{['th4']}
  11. Discussion and Conclusion

Key Result

Theorem 1.1

Assume that $u_0\in B^s_{p,r}(\mathbb{R}^3)$ with $(s,p,r)$ satisfying Then there exists a short time $T>0$ such that the Euler-Coriolis system CE admits a unique solution $u\in L^\infty([0,T];B^s_{p,r})$ satisfying that Furthermore, if $r<\infty$, the above solution $u\in C([0,T];B^s_{p,r})$ is continuous dependence in $B^s_{p,r}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1: B
  • ...and 13 more