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Global regularity of some axisymmetric, single-signed vorticity in any dimension

Deokwoo Lim

Abstract

We consider incompressible Euler equations in any dimension $ d\geq3 $ imposing axisymmetric symmetry without swirl. While the global regularity of smooth flows in this setting has been well-known in $ d=3 $, the same question in higher dimensions $ d\geq4 $ remains unsolved. Recently, global regularity for the case $ d=4 $ with some extra decay assumption on vorticity is obtained by proving global estimate of the radial velocity. Now we prove that the vorticity with single-sign and a similar decay assumption is globally regular for any $ d\geq4 $. This is due to pointwise decay estimate of radial velocity in sufficiently large radial distance, which depends on time. The result is of confinement type for support growth, which is going back to Marchioro [Comm. Math. Phys., 164 (1994) 507-524] and Iftimie--Sideris--Gamblin [Comm. Partial Differential Equations, 24 (1999) 1709-1730] for $ \mathbb{R}^{2} $. In particular, we follow the approach of Maffei--Marchioro [Rend. Sem. Mat. Univ. Padova, 105 (2001) 125-137] for $ d=3 $ so that we generalize the confinement into any dimension.

Global regularity of some axisymmetric, single-signed vorticity in any dimension

Abstract

We consider incompressible Euler equations in any dimension imposing axisymmetric symmetry without swirl. While the global regularity of smooth flows in this setting has been well-known in , the same question in higher dimensions remains unsolved. Recently, global regularity for the case with some extra decay assumption on vorticity is obtained by proving global estimate of the radial velocity. Now we prove that the vorticity with single-sign and a similar decay assumption is globally regular for any . This is due to pointwise decay estimate of radial velocity in sufficiently large radial distance, which depends on time. The result is of confinement type for support growth, which is going back to Marchioro [Comm. Math. Phys., 164 (1994) 507-524] and Iftimie--Sideris--Gamblin [Comm. Partial Differential Equations, 24 (1999) 1709-1730] for . In particular, we follow the approach of Maffei--Marchioro [Rend. Sem. Mat. Univ. Padova, 105 (2001) 125-137] for so that we generalize the confinement into any dimension.
Paper Structure (13 sections, 8 theorems, 120 equations)

This paper contains 13 sections, 8 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Axisymmetric, swirl-free Euler equations
  3. Main result
  4. Literature review and key ideas
  5. Support growth of a patch for any $d\geq3$
  6. The explicit form of $u^{r}$
  7. A time-independent pointwise estimate of $u^{r}$ for a patch
  8. Support growth of a single-signed vorticity for any $d\geq3$
  9. Refinement of the pointwise estimate of $u^{r}$
  10. Some technical lemmas
  11. Proof of the main result
  12. Proof of Proposition \ref{['prop_mafmar']}
  13. Proof of Theorem \ref{['thm:3']}

Key Result

Theorem 1.1

Let $d\geq3$, and assume that $\omega_{0}$ is single-signed, compactly supported in $\mathbb R^{d}$, and satisfies $r^{-(d-2)}\omega_{0}\in L^{\infty}(\mathbb R^{d})$. Then the corresponding solution $\omega$ of eq_vortformNd is global in time. In particular, for any $t\geq0$, it satisfies for some $C>0$ depending only on $d$, $\Vert r\omega_{0}\Vert_{L^{1}(\mathbb R^{d})}$, $\Vert r^{-(d-2)}\ome

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: CJLglobal22
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['prop_simpleur']}
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 8 more