Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

Diffuse interface surface tension models in an expanding flow

Pages: 387 – 418



Andrea Bertozzi (Department of Mathematics, University of California at Los Angeles)

Theodore Kolokolnikov (Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada)

Wangyi Liu (Department of Mathematics, University of California at Los Angeles)


We consider a diffusive interface surface tension model under compressible flow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field. These are two reduced models which show important properties of the full-scale surface tension model. 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 effective 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.


diffuse interface, surface tension, Cahn-Hilliard equation, numerical simulatio

2010 Mathematics Subject Classification

35B32, 35B50, 76T10

Published 14 October 2011