Unique solvability of the free-boundary Navier-Stokes equations with surface tension

TL;DR

This paper establishes the existence and uniqueness of solutions to the free-boundary Navier-Stokes equations with surface tension, using fixed-point theorems, energy methods, and regularity analysis.

Contribution

It introduces a novel approach combining fixed-point theorems and energy estimates to prove well-posedness of free-boundary Navier-Stokes with surface tension.

Findings

Proved existence and uniqueness of solutions.

Developed new spacetime inequalities for energy estimates.

Established regularity and a priori bounds for solutions.

Abstract

We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution 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 inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process.