Existence and uniqueness of global weak solutions to a   Cahn-Hilliard-Stokes-Darcy system for two phase incompressible flows in   karstic geometry
  We study the well-posedness of a coupled Cahn-Hilliard-Stokes-Darcy system which is a diffuse-interface model for essentially immiscible two phase incompressible flows with matched density in a karstic geometry. Existence of finite energy weak solution that is global in time is established in both 2D and 3D. Weak-strong uniqueness property of the weak solutions is provided as well. 
Daozhi Han, Xiaoming Wang, Hao WU
math.AP