Note on the nonlinear stability of stratified fluid equilibria

Abstract

We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate for finite-amplitude perturbation of the basic equilibrium state. the main idea of our approach is based on a special decomposition of the density perturbation, namely, we split the density perturbation in two parts, the first part depends on time but has zero initial value, the second one is in some sense time-independent (its L2-norm is time-independent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the total perturbation. In our approach we avoid the problem of a smooth extension of a locally convex function beyond its initial domian of definition that arises in applications of Arnold's method. Taking advantage of this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of problem appears, e.g., in atmospheric physics when symmetric flows of a stratified fluid are considered. As a result, we obtain a sufficient condition for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbations of arbitrary amplitude.