Wellposedness and Decaying Property of Viscous Surface Wave

TL;DR

This paper extends the analysis of viscous surface waves governed by Navier-Stokes equations, providing new techniques for wellposedness and decay properties in finite and infinite domains, advancing understanding of free boundary fluid dynamics.

Contribution

It generalizes local wellposedness results from small to arbitrary data and simplifies the proof of global wellposedness using innovative mathematical techniques.

Findings

Extended local wellposedness to arbitrary data

Provided simpler proof for global wellposedness

Developed new methods for energy and dissipation estimates

Abstract

In this paper, we consider an incompressible viscous flow without surface tension in a finite-depth domain of three dimensions, with free top boundary and fixed bottom boundary. This system is governed by a Naiver-Stokes equation in above moving domain and a transport equation for the top boundary. Traditionally, we consider this problem in Lagrangian coordinates with perturbed linear form. In the series of papers [1], [2] and [3], I. Tice and Y. Guo introduced a new framework using geometric structure in Eulerian coordinates 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. Also, we give a simpler proof for global wellposedness in infinite domain. Other than the geometric energy estimates, time-dependent Galerkin method, and interpolation estimate with Riesz potential and minimum count, which are introduced in these papers, we utilize three new techniques: (1) using \epsilon-Poisson integral to construct a diffeomorphism 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; (3) redefining the energy and dissipation to replace the original complicated bootstrapping argument to show interpolation estimate.