  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.
