Evidence of singularities for a family of contour dynamics equations

摘要

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < α ≤ 1. The limiting case α → 0 corresponds to 2D Euler equations, and α = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.