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

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

2002-12-09

arxiv.org/abs/math/0212116v2

Mathematics

Analysis of PDEs

73 pages; typos corrected; minor details added

Scientific paper

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.

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