On the blow up of a non-local transport equation in compact manifolds

TL;DR

This paper demonstrates finite time blow-up for certain non-local active scalar equations on compact Riemannian manifolds using a De Giorgi-type method, extending previous techniques to a broader geometric setting.

Contribution

It introduces a novel application of De Giorgi's method to establish blow-up in non-local equations on compact manifolds, expanding the understanding of singularity formation in these systems.

Findings

Finite time blow-up proven for specific non-local scalar equations

Extension of De Giorgi's method to Riemannian manifolds

Broader class of equations exhibiting singularity formation

Abstract

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol to deal with the one dimensional C\'ordoba-C\'ordoba-Fontelos equation and might be regarded as an instance of De Giorgi's method.