Stability of the 2D anisotropic Boussinesq equations with mixed partial dissipation

Abstract

Stability is an important property of solutions to many fluid flow equations. In this paper, we focus on the stability problem for two-dimensional Boussinesq equations with only vertical viscosity and horizontal thermal diffusion near a stratified, hydrostatic equilibrium in the domain [Formula: see text], where [Formula: see text] is a 1 D periodic box. We are mainly concerned with the H2-stability of global solutions when the initial data are closed to an equilibrium state and sufficiently small. Due to the lack of horizontal viscosity in velocity equations and vertical thermal diffusion in temperature equation, this problem seems to be not trivial and becomes particularly difficult. In order to achieve our main goal, we divide the velocity field and temperature into two parts: the horizontal average and the remaining oscillation part. Besides, we fully exploit the [Formula: see text]-norm of ∂xu2to help us dominate the nonlinear parts. Our result shows that with a small initial perturbation, there exists a unique global solution to the stated Boussinesq equations. In addition, the perturbation remains small for all time. In particular, our result also demonstrates the stabilizing effect of internal gravity waves, which are induced by the stratified background hydrostatic state.