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.

31 pages, 12 figures

References in corpus (1)

• Absence of splash singularities for SQG sharp fronts and the Muskat problem

Cited by in corpus (3)

• Finite time singularity formation for the modified SQG patch equation
• Generalized surface quasi-geostrophic equations with singular velocities
• The $\alpha$-SQG patch problem is illposed in $C^{2,\beta }$ and $W^{2,p}$