Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions
Fabian Bleitner, Elizabeth Carlson, Camilla Nobili
TL;DR
This work analyzes the 2D thermally non-diffusive Boussinesq system ($\mu=0$) with Navier-slip boundary conditions in a bounded domain, establishing uniform-in-time $H^2$-level regularity for the velocity and strong bounds for pressure and the active scalar. The authors derive sharp large-time asymptotics, showing $u(t)\to 0$ in $W^{1,q}$ for all $q<\infty$, while a pressure-velocity coupling remains controlled in the weak $L^2$ sense via a decomposition of $\theta e_2=\xi+\nabla\chi$, with $\xi$ divergence-free and curl-free. If $\theta e_2$ converges to a steady profile, the analysis yields strong convergence results: $\Delta u$ and $\nabla p-\theta e_2$ approach zero in $L^2$ (and the hydrostatic balance is recovered in the limit). A linear stability result in a periodic strip shows hydrostatic equilibrium is stable for $\beta>0$, consistent with vertical stable stratification. The paper extends prior results for free-slip and no-slip boundaries by enabling direct bounds on the pressure term under Navier-slip, sharpening the understanding of hydrostatic balance in thermally non-diffusive regimes.
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 $, $ θ= βx_2 + γ$, and $ p = \fracβ{2}x_2^2 + γx_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.], 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.