Asymptotic behaviour of the perturbation of a given mean flow

Abstract

We consider an oceanic domain included in R 3, in which there exist, at initial time, a current field V 0 and a temperature field θ 0. Perturbations V and θ of the velocity and the temperature are induced by a perturbation of the mean wind-stress. V and θ have to satisfy a non-linear problem of Navier-Stokes type. We prove the existence of the solution, for the variational problem, and give some results about uniqueness and regularity. In order to study the asymptotic behaviour of the perturbation, we introduce some operators, deduced from the Stokes operator. Their properties allow us to make a priori estimations, and to prove that, under some assumptions, the perturbations V( t) and θ( t) remain bounded as t → ∞. With stronger assumptions about the initial data, we can prove that the perturbation tends to 0 as t → ∞, and that the solution of the variational problem is a strong solution for every t ϵ[0, ∞[.