Global well-posedness of critical surface quasigeostrophic equation on the sphere

In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise inequality for fractional laplacians as in the work of Constantin and Vicol for the euclidean settingThis work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 and the MTM2011-2281 project of the MCINN (Spain

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

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