Global Well-Posedness and Sharp-Interface Limits of the Phase-Field Navier-Stokes Equations

Steve Shkoller, Mathematics, University of California, Davis

Phase-field models have been widely studied in many physical scenarios wherein phase transitions occur. The basic idea in this methodology is to use a scalar-valued function (the phase-field) to track the interface between two phases as its zero level-set and to simultaneously smooth over the traditional sharp (discontinuous) transition between the two phases, by "fattening" the interface to be \epsilon.B Surprisingly, this approach has not been widely used to study general two-phase Navier-Stokes flows; I shall begin the lecture by introducing a new phase-field Navier-Stokes equation (derived in collaboration with C. Liu) which models the motion of two fluids with differing viscosities and densities, and whose interface is governed by surface tension. This equation has global Leray-type weak solutions in 3D, and rigorously converges (in the sense of distributions) to solutions of the sharp-interface Navier-Stokes equations with interface motion goverened by the fluid velocity (as is traditional) plus the mean curvature of the interface as well, with a small term, called the mobility, in front of the mean curvature. In the limit of zero mobility, weak solutions of the traditional Navier-Stokes equations are obtained. In the course of the proof, I shall comment on the connections with level-set methods, and distance-function approaches to the problem.