Existence of solutions to a two-dimensional model for nonisothermal   two-phase flows of incompressible fluids
  We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. This model was recently introduced in a previous paper of ours, where we proved existence of weak solutions in three space dimensions. Here, we aim at studying the mathematical properties of the model in the two-dimensional case. In particular, we can show existence of global in time strong solutions. Moreover, we can admit slightly more general conditions on some material coefficients of the system. 
Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna
math.AP