Stabilization of the Gear-Grimshaw system with weak damping

TL;DR

This paper proves exponential stability for a coupled nonlinear Korteweg--de Vries system modeling internal gravity waves, using weak localized damping and advanced mathematical techniques.

Contribution

It introduces a novel stabilization approach for the Gear-Grimshaw system with very weak damping, establishing exponential decay of solutions.

Findings

Exponential stability of the system under weak damping.

Stability holds for nonlinear exponent in [1,4).

Uses Compactness--Uniqueness and unique continuation methods.

Abstract

The aim of this work is to consider the internal stabilization of a nonlinear coupled system of two Korteweg--de Vries equations in a finite interval under the effect of a very weak localized damping. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Considering feedback controls laws and using Compactness--Uniqueness Argument, which reduce the problem to use a unique continuation property, we establish the exponential stability of the weak solutions when the exponent in the nonlinear term ranges over the interval $[1,4)$.