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

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 [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi’s method.