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On global regularity of some bi-rotational Euler flows in $\mathbb{R}^{4}$

Kyudong Choi, In-Jee Jeong, Deokwoo Lim

Abstract

In this paper, we consider incompressible Euler flows in $ \mathbb{R}^{4} $ under bi-rotational symmetry, namely solutions that are invariant under rotations in $\mathbb{R}^{4}$ fixing either the first two or last two axes. With the additional swirl-free assumption, our first main result gives local wellposedness of Yudovich-type solutions, extending the work of Danchin [Uspekhi Mat. Nauk 62(2007), no.3, 73-94] for axisymmetric flows in $\mathbb{R}^{3}$. The second main result establishes global wellposedness under additional decay conditions near the axes and at infinity. This in particular gives global regularity of $C^{\infty}$ smooth and decaying Euler flows in $\mathbb{R}^{4}$ subject to bi-rotational symmetry without swirl.

On global regularity of some bi-rotational Euler flows in $\mathbb{R}^{4}$

Abstract

In this paper, we consider incompressible Euler flows in under bi-rotational symmetry, namely solutions that are invariant under rotations in fixing either the first two or last two axes. With the additional swirl-free assumption, our first main result gives local wellposedness of Yudovich-type solutions, extending the work of Danchin [Uspekhi Mat. Nauk 62(2007), no.3, 73-94] for axisymmetric flows in . The second main result establishes global wellposedness under additional decay conditions near the axes and at infinity. This in particular gives global regularity of smooth and decaying Euler flows in subject to bi-rotational symmetry without swirl.
Paper Structure (23 sections, 11 theorems, 158 equations, 2 figures)

This paper contains 23 sections, 11 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The system for bi-rotationally symmetric solutions
  3. Main results
  4. Key ideas
  5. Literature review
  6. Axisymmetric solutions in three dimensions
  7. Axisymmetric Euler equations in higher dimensions
  8. The lake equation and the vortex membrane equation
  9. Comparison with 2D and 3D Euler equations
  10. Outline of the paper
  11. Preliminaries
  12. Lorentz spaces
  13. Vorticity formulation of the Euler equations
  14. Vorticity tensor in the bi-rotational geometry and derivation of the Biot--Savart law
  15. Local regularity of $w$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $w_{0}$ satisfy $w_{0} \in (L^{4,1}\cap L^\infty)(\mathbb R^4)$ and Then, there exist $T>0$ and a unique solution to eq_Eulereq satisfying $w \in L^\infty(0,T ;(L^{4,1}\cap L^\infty)(\mathbb R^4))$ and

Figures (2)

  • Figure 1: Overall flow pattern for signed vorticity: $\mathbb R^{2}$ under odd-odd symmetry (left), $\mathbb R^{3}$ under axisymmetry and odd symmetry in $z$ (center), $\mathbb R^{4}$ under bi-rotational symmetry (right).
  • Figure 2: Overall flow pattern for signed vorticity under an additional odd symmetry

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4: Global well-posedness for smooth and decaying data
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lem:4D-Biot-Savart']}
  • Lemma 3.1
  • proof
  • ...and 14 more