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

TL;DR

This paper investigates the self-similar collapse behavior of the alpha-patches problem through numerical methods, revealing complex blow-up scenarios and the role of exact solutions in initial data evolution.

Contribution

It introduces a numerical algorithm for the alpha-patches problem in self-similar variables and analyzes the stability of self-similar solutions in collapse dynamics.

Findings

Exact self-similar solutions influence collapse outcomes.

Perturbations around solutions can lead to blow-up or stability.

Numerical simulations suggest complex blow-up scenarios near rescaled profiles.

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.