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

Abstract

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 $t\to \infty$, $\Vert u{\Vert }_{W^{1,p}}\to 0$, 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.], which focus on free-slip boundary conditions, as well as recent studies [Aydın, Kukavica, Ziane; Aydın, Jayanti] 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.

35 pages

References in corpus (3)

• Oscillatory convection and limitations of the Boussinesq approximation
• Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries
• Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains