Global Axisymmetric Euler Flows with Rotation

TL;DR

This paper constructs global axisymmetric solutions to the 3D Euler equations with rotation, demonstrating linear scattering due to dispersive effects and introducing a symmetry-based analytical framework.

Contribution

It provides the first class of global solutions near rigid body rotation with non-vanishing swirl, utilizing a novel symmetry and dispersive analysis framework.

Findings

Solutions scatter linearly due to rotation-induced dispersion

Framework captures anisotropic dispersive mechanisms

Propagates sharp decay bounds for global flow construction

Abstract

We construct a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.