Local Wellposedness of Viscous Surface Wave without Surface Tension

TL;DR

This paper proves local well-posedness for viscous surface wave equations without surface tension in three dimensions, extending previous results from small to arbitrary initial data using novel geometric and analytical techniques.

Contribution

It extends the local well-posedness results for viscous surface waves without surface tension from small to arbitrary data by employing new geometric and analytical methods.

Findings

Established local well-posedness for arbitrary data.

Developed a new transform between fixed and moving domains.

Proved a comparison result for steady Navier-Stokes equations.

Abstract

We consider an incompressible viscous flow without surface tension in a finite- depth domain of three dimension, with free top boundary. This system is governed by a Naiver-Stokes equation in a moving domain and a transport equation for the top boundary. Traditionally, we consider this problem in Lagrangian coordinate and perturbed linear form. In [1], I. Tice and Y. Guo introduced a new framework using geometric structure in Eulerian coordinate to study both local and global wellposedness of this system. Following this path, we extend their result in local wellposedness from small data case to arbitrary data case. Other than the geometric energy estimate and time-dependent Galerkin method introduced in [1], we utilize a few new techniques: (1) using parameterized Poisson integral to construct a nontrivial transform between fixed domain and moving domain; (2) using bootstrapping argument to prove a comparison result for steady Navier-Stokes equation for arbitrary data of free surface.