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The goal of this paper is to study the large-time bahaviour of a buoyancydriven fluid without thermal diffusion and Navier-slip boundary conditions in abounded domain with Lipschitz-continuous second derivatives. After showingglobal well-posedness and regularity of classical solutions, we study theirlarge-time asymptotics. Specifically we prove that, in suitable norms, thesolutions converge to the hydrostatic equilibrium. Moreover, we prove linearstability for the hydrostatic equilibrium when the temperature is an increasingaffine function of the height, i.e. the temperature is vertically stablystratified. This work is inspired by results in [Doe+18] for free-slip boundaryconditions.

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