Stabilization of the water-wave equations with surface tension

TL;DR

This paper demonstrates that applying a specific external pressure on a portion of the free surface can exponentially stabilize the water-wave equations with surface tension, leading to energy decay over time.

Contribution

It introduces a novel stabilization method using external pressure proportional to the normal velocity component, ensuring exponential energy decay in water-wave models.

Findings

Energy decays exponentially with the proposed control.

External pressure based on normal velocity achieves stabilization.

Method applies to water-wave equations with surface tension.

Abstract

This paper is devoted to the stabilization of the water-wave equations with surface tension through of an external pressure acting on a small part of the free surface. It is proved that the energy decays to zero exponentially in time, provided that the external pressure is given by the normal component of the velocity at the free surface multiplied by an appropriate cut-off function.