University of California, Davis
The incompressible Navier-Stokes equations (NSE) on time-dependent domains arise in the study of either a single fluid with a free-surface, or in the context of multi-phase flows wherein the motion of two or more immiscible fluids is considered. In the presence of surface tension, the mathematical analysis (as well as the numerical computation) is a challenging task, because the mean curvature vector, given in the surface tension term, appears to induce too much derivative loss on the boundary; in fact, Newton iteration will fail to converge, and other iteration schemes must be constructed.
I will describe the analysis of surface-tension driven interface motion in both the short-time and long-time regimes. For short-time well-posedness of the NSE, I will present a technique, based on new types of energy laws of the linearized system. For long-time simulations, a generalization of the NSE is required to make sense of the mean curvature vector at the point of singularity. While viscosity solution techniques for the NSE have been employed when surface tension is assumed to be zero, with surface tension present such techniques are not known to hold. I will describe a phase-field model that fattens-up the sharp interface of the NSE and has long-time weak solutions. I will then explain how solutions of this phase-field model weakly converge to solutions of the NSE, as long as the NSE solutions exist.