  This paper studies the dynamical evolution of the alpha-patches problem
expressed in self-similar variables. A numerical algorithm is proposed and
these equations are numerically explored. Several benchmarks of the code are
discussed throughout the paper. Exact self-similar solutions are described and
are found to play a role in separating collapsing from non-collapsing initial
data: small perturbations around this solution blow up while others do not.
Numerical simulations performed near convergent rescaled profiles, such as
those described by Cordoba et al. in [12], indicate the absence of a stationary
graph in the neighborhood of the rescaled profiles and suggest a more complex
scenario for blow up.
