A Morawetz inequality for gravity-capillary water waves at low Bond number

TL;DR

This paper proves a Morawetz inequality for 2D gravity-capillary water waves, demonstrating global-in-time estimates with surface tension effects, uniform in physical limits, and showing a smoothing effect for nonlinear waves.

Contribution

It extends previous work by incorporating surface tension and establishing a global Morawetz inequality with regularity gain for nonlinear water waves.

Findings

Global Morawetz inequality for gravity-capillary waves

Uniform estimates in depth and surface tension limits

Surface tension induces a smoothing effect in solutions

Abstract

This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.