Unique Solvability of the Free-boundary Navier-stokes Equations with Surface Tension

We prove the existence and uniqueness of solutions to the timedependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem that requires the analysis of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime estimates which allow us to find a unique weak solution to this linear problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the topological fixed-point theorem.