On the Effects of Advection and Vortex Stretching

TL;DR

This paper proves finite-time singularity formation in certain vortex models, demonstrating the complex interplay of stabilizing and destabilizing effects, and explores conditions leading to singularity or global regularity.

Contribution

It establishes finite-time singularities for De Gregorio's model and Okamoto-Sakajo-Wunsch models in new parameter ranges, advancing understanding of vortex dynamics.

Findings

Finite-time singularity for De Gregorio's model in specific function spaces.

Singularity formation from smooth initial data in Okamoto-Sakajo-Wunsch models.

Finite-energy solutions of the surface quasi-geostrophic equation can develop singularities.

Abstract

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^\alpha(\mathbb{R})$ vorticities for some $\alpha>0$ and $p<\infty$. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are $C^\alpha$-regular. One of the difficulties in the models we consider is that there are competing \emph{nonlocal} stabilizing effects (advection) and destabilizing effects (vortex stretching) which are of the same size in terms of scaling. Hence, it is difficult to establish the domination of one effect over the other without having strong control of the solution. We conjecture that strong solutions to the De Gregorio model exhibit the following behavior: for each $0<\alpha<1$ there exists an initial $\omega_0\in C^\alpha(\mathbb{R})$ which is compactly supported for which the solution becomes singular in finite-time; on the other hand, solutions to De Gregorio's equation are global whenever $\omega_0\in L^p\cap C^{1}(\mathbb{R})$ for some $p<\infty$. Such a dichotomy seems to be a genuinely non-linear effect which cannot be explained merely by scaling considerations since $C^\alpha$ spaces are scaling subcritical for each $\alpha>0$.