This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where as \(t \rightarrow \infty \) , \(\Vert u\Vert _{W^{1,p}} \rightarrow 0\) , and hydrostatic balance is achieved in the weak topology of \(L^2\) . Furthermore, we identify the necessary conditions under which stable stratification and hydrostatic balance can be achieved in the strong topology as time approaches infinity. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by \(u = 0\) , \(\theta = \beta x_2 + \gamma \) and \(p = \frac{\beta }{2}x_2^2 + \gamma x_2 + \delta \) . In a periodic strip, we establish the linear stability of this state for \(\beta > 0\) , indicating that the temperature is vertically stably stratified. This work builds upon the results in Doering et al. (Phys D Nonlinear Phenom 376–377:144-159, 2018), which focus on free-slip boundary conditions, as well as recent studies (Aydın and Jayanti in Fractional regularity, global persistence and asymptotic properties of the Boussinesq equations on bounded domains, https://arxiv.org/abs/2403.12509, 2024; Aydın et al. in On asymptotic properties of the Boussinesq equations, https://arxiv.org/abs/2304.00481, 2023) that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions.
