  We study the asymptotic limit of diffused surface energy in the van der
Waals--Cahn--Hillard theory when an advection term is added and the energy is
uniformly bounded. We prove that the limit interface is an integral varifold
and the generalized mean curvature vector is determined by the advection term.
As the application, a prescribed mean curvature problem is solved using the
min-max method.
