Large time behaviour of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

The goal of this paper is to study the large-time behaviour of a buoyancy driven fluid without thermal diffusion and Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After showing improved regularity of classical solutions, we study their large-time asymptotics. Specifically we prove that, in suitable norms, the solution converges to the hydrostatic equilibrium. Moreover, we prove linear stability for the hydrostatic equilibrium when the temperature is an increasing affine function of the height, i.e. the temperature is vertically stably stratified. This work is inspired by results in [Doe+18] for free-slip boundary conditions.