On the stabilizing effect of rotation in the 3d Euler equations

TL;DR

This paper demonstrates that constant rotation in the 3D Euler equations stabilizes solutions over long times and induces dispersion, regardless of the rotation speed, through an anisotropic analytical framework.

Contribution

It introduces a novel anisotropic framework to analyze the stabilizing and dispersive effects of rotation in the 3D Euler equations for small axisymmetric initial data.

Findings

Solutions exist for at least ^{-M} time for small initial data.

Rotation induces dispersive decay in the solutions.

Stabilizing effect is independent of rotation speed.

Abstract

While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in $\mathbb{R}^3$ with a fixed speed of rotation. We show that for any $M>0$, axisymmetric initial data of sufficiently small size $\varepsilon$ lead to solutions that exist for a long time at least $\varepsilon^{-M}$ and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.