Exact nonstationary solutions of the Euler equations

The Euler equations of ideal fluid mechanics are notoriously difficult to solve, except in the case of stationary solutions. Only a few exact nonstationary solutions are known, including Rossby-Haurwitz waves on the sphere that appear in weather modeling and Kelvin waves in a rotating cylinder. Here I will describe a generalization of these solutions that gives new formulas for exact solutions in two and three dimensions, which are perturbations around a rigid motion for which the linearized solution happens to be an exact solution. In two dimensions these can only occur in very special geometries, while in three dimensions there are many more; I will describe a partial classification of them involving those Riemannian geometries with special families of isometries. This is joint work with Patrick Heslin.