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

TL;DR

This paper studies the long-term behavior of 2D buoyancy-driven fluid flows without thermal diffusion under Navier-slip conditions, showing convergence to hydrostatic balance and analyzing stability of steady states.

Contribution

It provides new results on the large-time asymptotics and stability of solutions to the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions, including direct pressure bounds.

Findings

Solutions converge to hydrostatic balance as time approaches infinity.

Stable stratification occurs under certain conditions, with stability proven for specific steady states.

Pressure can be directly bounded due to 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 $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, 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.], which focus on free-slip boundary conditions, as well as recent studies [Ayd{\i}n, Kukavica, Ziane; Ayd{\i}n, 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.