**The Navier-Stokes Equations with Moving Interfaces and Surface
Tension**

Steve Shkoller

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.

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