Evidence of singularities for a family of contour dynamics equations

Diego Córdoba; Marco A Fontelos; Ana M Mancho; Jose L Rodrigo

doi:10.1073/pnas.0501977102

Evidence of singularities for a family of contour dynamics equations

Proc Natl Acad Sci U S A. 2005 Apr 26;102(17):5949-52. doi: 10.1073/pnas.0501977102. Epub 2005 Apr 18.

Authors

Diego Córdoba¹, Marco A Fontelos, Ana M Mancho, Jose L Rodrigo

Affiliation

¹ Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, C/Serrano 123, Madrid 28006, Spain. dcg@imaff.cfmac.csic.es

Abstract

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 < alpha </= 1. The limiting case alpha --> 0 corresponds to 2D Euler equations, and alpha = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.