Numerical studies on the self-similar collapse of the alpha-patches problem

Abstract

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.