Well-posedness of a water wave model with viscous effects

TL;DR

This paper derives a nonlocal fourth order wave equation modeling viscous water waves in deep water, including surface tension effects, and proves its well-posedness in Sobolev spaces, advancing the mathematical understanding of viscous free surface flows.

Contribution

It introduces a new nonlocal wave model for viscous water waves and establishes its well-posedness, extending prior models to include viscosity and surface tension effects.

Findings

Derived a nonlocal fourth order wave equation for viscous water waves

Proved well-posedness of the model in Sobolev spaces

Includes effects of surface tension in the model

Abstract

Starting from the paper by Dias, Dyachenko and Zakharov (\emph{Physics Letters A, 2008}) on viscous water waves, we derive a model that describes water waves with viscosity moving in deep water with or without surface tension effects. This equation takes the form of a nonlocal fourth order wave equation and retains the main contributions to the dynamics of the free surface. Then, we prove the well-posedness in Sobolev spaces of such equation.