Asymptotic Stability of Equilibrium States with Variable Temperature Gradient to the Boussinesq System Without Thermal Conduction

We study the asymptotic stability of equilibrium states with positive (and variable) temperature gradient to the Boussinesq system without thermal conduction in the strip domain ℝ² × (0, 1). It is shown that a unique global-intime solution exists if the initial data is close enough to such an equilibrium state with suitable boundary conditions. Moreover, as time goes to infinity, the solution converges to the corresponding equilibrium state with explicit decay rates. Such a result reflects the well-known Rayleigh–Taylor stability phenomenon in the fluid motion.