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
→
∞
,
‖
u
‖
W
1
,
p
→
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
,
θ
=
β
x
2
+
γ
and
p
=
β
2
x
2
2
+
γ
x
2
+
δ
. In a periodic strip, we establish the linear stability of this state for
β
>
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.