# The nonlinear stability regime of the viscous Faraday wave problem

## TL;DR

This paper investigates the nonlinear stability of viscous Faraday waves, demonstrating that small perturbations decay over time under certain conditions, with or without surface tension, in a fluid layer above an oscillating boundary.

## Contribution

It establishes the existence of parameter regimes where flat equilibrium solutions are asymptotically stable for viscous fluids with free surfaces, extending understanding of Faraday wave dynamics.

## Key findings

- Small perturbations decay to equilibrium over time.
- Stability results hold with and without surface tension.
- Quantitative decay rates are identified.

## Abstract

This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium at an identified quantitative rate.