Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system

Abstract

We consider a coupled system of Kuramoto-Sivashinsky equations depending on a suitable parameter ν > 0 and study its asymptotic behavior for t large, as ν → 0. Introducing appropriate boundary conditions we show that the energy of the solutions decays exponentially uniformly with respect to the parameter ν. In the limit, as ν → 0, we obtain a coupled system of Korteweg-de Vries equations known to describe strong interactions of two long internal gravity waves in a stratified fluid for which the energy tends to zero exponentially as well. The decay fails when the length of the space interval L lies in a set of critical lengths.