Global existence for some transport equations with nonlocal velocity
  In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of an one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and $n$ dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation. 
Hantaek Bae, Rafael Granero-Belinch\'on
math.AP