Di use interface surface tension models in an expanding ow

We consider a di usive interface surface tension model under compressible ow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity eld. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more e ective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle.