Global well-posedness for a transport equation with non-local velocity and critical diffusion

Abstract

This paper is concerned with a one dimensional transport equationwith a non-local velocity and critical diffusion. In [Ann. ofMath. 162 (2005), 1377--1389], A. Córdoba, D.Córdoba and Marco A. Fontelos showed the finite timesingularities for a family of initial data without diffusion,global existence with subcritical diffusion and the globalexistence for small initial data with critical diffusion. We provethe global existence with critical diffusion, removing thesmallness constraint on the initial data.