Apparaît dans la collection : 2026 - T2 - WS3 - Idealised mathematical models for geophysical flows

Exact solutions to the two dimensional Euler's equations, on Euclidean and non-Euclidean surfaces, are presented in Lagrangian coordinates. These solutions arise due to a particle relabeling invariance, a subset of which, associated with particle label rotations, are shown to transform time independent solutions to time dependent solutions by these infinitesimal canonical transformations. The associated compatibility conditions of these maps restrict the label dependence to be harmonic maps from cartesian label space to these two dimensional surfaces, connecting the rotational relabelling symmetry with harmonic maps. Using a frame equation approach on the sphere, harmonic maps from the plane to the sphere are associated with a negative sinh Laplace equation and the associated family of these maps, which rotate the Hopf differential, are shown to generate the time evolution. Simpler solutions with label dependent rotations are also presented.

Co-authors: Rick Salmon