Stability and large‐time behavior of the 3D Boussinesq equations with mixed partial kinematic viscosity and thermal diffusivity near one equilibrium

Abstract

In this paper, we study the stability problem and large‐time behavior for the 3D Boussinesq equations with mixed partial kinematic viscosity and thermal diffusivity near one equilibrium, i.e., with (where is chosen to be large, e.g., satisfying the well‐known Miles‐Howard criterion). These problems can be extremely difficult due to the equations lack of full dissipation. Our results are twofold. In the first place, we establish the global and stability for the nonlinear system with seven different partially dissipated systems of Boussinesq equations. In the second place, we derive precise large‐time behavior for the linearized system.