The Kelvin-Helmholtz Instability of Momentum Sheets in the Euler Equations for Planar Diffeomorphisms

The Euler equations that describe geodesics on the group of diffeomorphisms of the plane admit singular solutions in which the momentum is concentrated on curves, the so-called momentum sheets analogous to vortex sheets in the Euler fluid equations. We study the stability of straight and circular momentum sheets for a large family of metrics. We prove that straight sheets moving normally to themselves under an H metric, corresponding to peakons for the one-dimensional (1D) Camassa– Holm equation, are linearly stable in Eulerian coordinates, suffering only a weak instability of Lagrangian particle paths, while most other cases are unstable but well-posed. Expanding circular sheets are algebraically unstable for all metrics. The evolution of the instabilities are followed numerically, illustrating several typical dynamical phenomena of momentum sheets.