Cusp Formation in Vortex Patches

Cohen and Danchin (2000) numerically conjectured that any initial vortex patch with acute corners instantaneously develops into a cusp structure of order one with logarithmic sharpness. In this talk, based on recent joint work with T. Elgindi, we present a complete proof of this conjecture. Our main observation is that the dynamics near the singular points reduce to a nonlinear singular fourth-order ODE that captures the cusp formation mechanism.