Stability and decay rates for a variant of the 2D Boussinesq-Binard system

Abstract

This paper investigates the stability and large-time behavior of perturbations near the trivial solution to a variant of the 2D Boussinesq-Binard system. This system does not involve thermal diffusion. Our research is partially motivated by a recent work [C.R. Doering, J. Wu, K. Zhao, and X. Zheng, Phys. D, 376/377:144[159, 2018] on the stability and large-time behavior of solutions near the hydrostatic balance concerning the 2D Boussinesq system. Due to the lack of thermal diffusion, these stability problems are difficult. The energy method and classical approaches are no longer effective in dealing with these partially dissipated systems. This paper presents a new approach that takes into account the special structure of the linearized system. The linearized parts of the vorticity equation and the temperature equation both obey a degenerate damped wave-type equation. By representing the nonlinear system in an integral form and carefully crafting the functional setting for the initial data and solution spaces, we are able to establish the long-term stability and global (in time) existence and uniqueness of smooth solutions. Simultaneously, we also obtain exact decay rates for various derivatives of the perturbations.