Global well-posedness and decay for viscous water wave models

TL;DR

This paper introduces a new asymptotic model for viscous water waves, proving its global well-posedness and decay properties, thus advancing understanding of viscous effects in water wave dynamics.

Contribution

It derives and analyzes a novel viscous water wave model, establishing global well-posedness and decay results for unidirectional and bidirectional wave equations.

Findings

Proved global well-posedness in Sobolev spaces for the models.

Established decay properties of the solutions.

Extended analysis to bidirectional water wave models with viscosity.

Abstract

The motion of the free surface of an incompressible fluid is a very active research area. Most of these works examine the case of an inviscid fluid. However, in several practical applications, there are instances where the viscous damping needs to be considered. In this paper we derive and study a new asymptotic model for the motion of unidirectional viscous water waves. In particular, we establish the global well-posedness in Sobolev spaces. Furthermore, we also establish the global well-posedness and decay of a fourth order PDE modelling bidirectional water waves with viscosity moving in deep water with or without surface tension effects.