Trochoidal Solutions to the Incompressible Two-Dimensional Euler Equations

Within the Lagrangian framework we present an approach yielding some explicit solutions to the incompressible two-dimensional Euler equations, generalizing the celebrated Gerstner flow. The solutions so obtained, for which explicit formulas of each particle trajectory are provided, represent either flows in domains with a rigid boundary or free-surface flows for a fluid of infinite depth. For some of these solutions the trajectories are epitrochoids or hypotrochoids. Possibilities for obtaining further flows of this type are indicated.