Large-Time Behavior of Solutions to Darcy–Boussinesq Equations with Non-Vanishing Scalar Acceleration Coefficient

Abstract

We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form $(\mathbf{u},\rho ,p)=(\mathbf{0},{\rho }_s,p_s)$, where ${\rho }_s$ is characterised by the hydrostatic balance $\nabla p_s=-{\rho }_s\nabla \mathrm{\Psi }$. Second, we prove that the steady-state solution satisfying $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$ is linearly stable provided that $\delta (x,y)<{\delta }_0<0$, while the system exhibits Rayleigh–Taylor instability if $\mathrm{\Psi }=gy$, ${\rho }_s={\delta }_{0}g$ and ${\delta }_0>0$. Finally, despite the inherent Rayleigh–Taylor instability that may trigger exponential growth in time, we prove that for any sufficiently regular initial data, the solutions of the system asymptotically converge towards the vicinity of a steady-state solution, where the velocity field is zero, and the new state is determined by hydrostatic balance. This work advances porous media modeling for geophysical and engineering applications, emphasizing the critical interplay of gravity, inertia, and boundary conditions.

1. Introduction

The investigation of hydrodynamical stability constitutes a crucial and central area within the vast realm of nonlinear sciences. It aims to enhance our understanding of the dynamics of fluid flows and to devise effective techniques and instruments to manage these flows in real-world applications. Given its significant importance in revealing the complex characteristics of fluid flows, an extensive and continuously expanding collection of research has been published, focusing on the mathematical analysis of hydrodynamical stability. For more in-depth studies and additional references, the reader is directed to the works [1,2,3,4,5,6,7,8,9,10,11] and the references cited therein. A key method in the study of hydrodynamical stability is to examine the stability of steady-state solutions and to explore the large-time behavior of the partial differential equations that describe fluid motion. This method offers vital insight into the fundamental principles governing fluid dynamics.

In this paper, we investigate the large-time behavior of solutions to Darcy–Boussinesq equations with a scalar acceleration coefficient. The Boussinesq equations are given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+{\displaystyle \frac{\mu }{K}}\mathbf{u}=-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ {\displaystyle \frac{\partial \rho }{\partial t}}+(\mathbf{u}·\nabla )\rho =0,\hfill \\ \nabla ·\mathbf{u}=0.\hfill \end{array}\right.\end{array}$$

(1)

where $\mathbf{u}={(u,v)}^T$ is the velocity field, $\rho$ is the density, $\mathrm{\Psi }$ is the gravity potential function and $C_a$ is called the scalar acceleration coefficient, which depends sensitively on the geometry of the porous medium and is determined mainly by the nature of the pore tubes with the largest cross-sections. The first equation of (1) is a variant version of Darcy’s law formulated by including an extra inertial term proportional to ${\displaystyle \frac{\partial \mathbf{u}}{\partial t}}$; see (6.6) in the book [12]. The second equation of (1) expresses the equation of continuity. For $C_a=0$, the system (1) is reduced to the standard Darcy–Boussinesq equations, which combine the effects of porous media flow and buoyancy-driven convection, play a crucial role in modeling geophysical and engineering phenomena, such as groundwater flow, oil reservoir dynamics, and thermal convection in porous media [12,13,14,15,16]. These equations extend the classical Boussinesq system by incorporating Darcy’s law, which describes the flow of fluids through porous media, thereby providing a more accurate representation of fluid dynamics in heterogeneous environments [16]. Despite their importance, the mathematical analysis of the Darcy–Boussinesq equations remains challenging due to the interplay between porous media effects, buoyancy forces, and the complex boundary conditions that often arise in practical scenarios.

At present, numerous mathematicians have also investigated the stability of steady-state solutions to Darcy–Boussinesq equations with a vanishing scalar acceleration coefficient ($C_a=0$) and constant-valued gravity, $\nabla \mathrm{\Psi }={(0,g)}^T$, the large-time asymptotic behavior of these equations, and the impact of various physical parameters on the overall dynamics. For example, the regularity and stability of the Darcy–Boussinesq system under different boundary conditions and with the influence of thermal and mechanical anisotropy have been explored in several works [17,18,19,20,21,22,23,24,25]. However, many questions remain open, particularly regarding the global regularity of solutions [26,27,28], the precise nature of instabilities [29], and the large-time convergence properties in the presence of partial dissipation or anisotropic diffusion [30].

Based on previous research, we studied the large-time behavior of solutions to Darcy–Boussinesq equations with $C_a\ne 0$, which model buoyancy-driven flows in porous media under spatially varying gravity and inertial effects. By integrating spectral analysis, energy estimates, and asymptotic theory, we resolve long-standing questions about stability, convergence, and the interplay between inertia and buoyancy. The findings not only deepen theoretical understanding but also provide actionable guidelines for engineering systems where gravity and inertia coexist.

1.1. Steady-State Solution

In practical scenarios, fluid flows generally take place within bounded regions and are influenced by boundary conditions, leading to the formulation of boundary value problems. Within the realm of geophysical and astrophysical fluid dynamics, when analyzing fluid motions governed by the Boussinesq approximation within a circular domain, it is more appropriate from a physical standpoint to replace the gravity term $-\rho /{\rho }^{*}{(0,g)}^T$ by $-\rho \nabla \mathrm{\Psi }/{\rho }^{*}$. Here, $\mathrm{\Psi }$ represents a potential function that characterizes the gravitational force. This adjustment provides a more accurate representation of the effects of spatially non-uniform gravitational forces.

When studying equations that model the motion of a fluid, having a good understanding of the exact steady-state solutions of the system is often helpful. The steady-state solutions of (1) are determined by the following equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\mu }{K}}\mathbf{u}=-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ (\mathbf{u}·\nabla )\rho =0,\hfill \\ \nabla ·\mathbf{u}=0,\hfill \end{array}\right.\end{array}$$

(2)

subject to the following boundary condition:

$$\begin{array}{c}\hfill {\mathbf{u}·\mathbf{n}|}_{\partial \mathrm{\Omega }}=0,\end{array}$$

(3)

where $\mathrm{\Omega }$ denotes a bounded domain with a smooth boundary, and the same applies below.

We declare that ${\parallel \mathbf{u}\parallel }_{L^2}$ is the abbreviation ${\parallel u\parallel }_{L^2\left(\mathrm{\Omega }\right)}$ that denotes the $L^2$ norm of $\mathbf{u}$ over the domain $\mathrm{\Omega }$ in this paper, and other forms are similar.

Lemma 1.

Let $(\mathbf{u},\rho )$ be a solution of Equation (2) subject to boundary condition (3); we then have $\mathbf{u}=0$, and ρ is determined by the following equation:

$$\begin{array}{c}\hfill {\nabla }^{\perp }\rho ·\nabla \mathrm{\Psi }=0.\end{array}$$

(4)

Proof. 

By multiplying the first equation of (2) by $\mathbf{u}$, after an integration by parts, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{\mu }{K}}{\left|\right|\mathbf{u}\left|\right|}_{L^2}^2=-{\int }_{\mathrm{\Omega }}\rho \mathbf{u}·\nabla \mathrm{\Psi }dxdy.\end{array}$$

(5)

By multiplying the second equation of (2) by $\mathrm{\Psi }$, after an integration by parts, one also obtains

$$\begin{array}{c}\hfill 0=-{\int }_{\mathrm{\Omega }}\rho (\mathbf{u}·\nabla )\mathrm{\Psi }dxdy.\end{array}$$

(6)

Relation (5) gives ${\left|\right|\mathbf{u}\left|\right|}_{L^2}=0$. □

Remark 1.

Suppose $\mathrm{\Delta }\mathrm{\Psi }=0$, for any bounded and smooth function $\delta (x,y)$. Let ${\rho }_s$ be a function such that $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$, and $p_s$ be a function such that $\mathrm{\Delta }{p}_s=-\delta (x,y){\left|\nabla \mathrm{\Psi }\right|}^2$; then $(\mathbf{u},\rho ,p)=(\mathbf{0},{\rho }_s,p_s)$ are a family of steady-state solutions to (1).

Generally speaking, steady-state solutions are potentially asymptotic states that could, to some extent, reveal the large-time behavior of a system modeled by partial differential equations (PDEs). Lemma 1 says that system (1) only allows steady-state solutions that are states of hydrostatic equilibria. This fact naturally prompts us to consider whether any solution of system (1) around any state of hydrostatic equilibrium $(\mathbf{0},{\rho }_s)$ satisfying $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$ and with sufficiently regular initial values converges to some stable state of hydrostatic equilibrium determined by (4). To this end, we consider the perturbation of any smooth steady-state $({\mathbf{u}}_s,{\rho }_s)=(\mathbf{0},{\rho }_s)$. This perturbation is modeled by the following Darcy–Boussinesq equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+{\displaystyle \frac{\mu }{K}}\mathbf{u}=-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ {\displaystyle \frac{\partial \rho }{\partial t}}+\delta (\mathbf{u}·\nabla )\mathrm{\Psi }=-(\mathbf{u}·\nabla )\rho ,\hfill \\ \nabla ·\mathbf{u}=0,\hfill \end{array}\right.\end{array}$$

(7)

subject to boundary condition (3) and with the following initial data:

$$\begin{array}{cc}\hfill & {\mathbf{u}|}_{t=0}={\mathbf{u}}_0{,\rho |}_{t=0}={\rho }_0.\hfill \end{array}$$

(8)

For system (1) with non-constant $\nabla \mathrm{\Psi }$ or even time-dependent $\nabla \mathrm{\Psi }$, the global asymptotic dynamics around general hydrostatic equilibria (stable and unstable equilibria) remain an unresolved issue. This article aims to address this issue.

The main results of this paper are summarized in the following theorems.

1.2. Main Results

Theorem 1.

Assume that $\delta (x,y)<{\delta }_0<0$ and $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$. For the steady-state solution $(\mathbf{0},{\rho }_s)$ where $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$, it is linearly stable. More precisely, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \underset{t\to \infty }{lim}{\left|\right|\mathbf{u}\left|\right|}_{L^2}^2=0,\underset{t\to \infty }{lim}{{\left|\right|\mathbf{u}\left|\right|}_t}_{L^2}^2=0,\hfill \\ \hfill & \underset{t\to \infty }{lim}{\left|\right|{\rho }_t\left|\right|}_{L^2}^2=0,\underset{t\to \infty }{lim}{\left|\right|\nabla p+\rho \nabla \mathrm{\Psi }\left|\right|}_{L^2}^2=0.\hfill \end{array}\end{array}$$

(9)

In particular, if $\mathrm{\Psi }=gy$, then the steady-state solution $(\mathbf{0},{\delta }_{0}g)$ is linearly unstable if ${\delta }_0>0$.

Theorem 1 is obtained by Lemma 2 and Proposition 1.

Theorem 2.

Assume $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$. For the classical solution $(\mathbf{u},\rho )$ of Equation (7) subject to boundary condition (3), with the initial data $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)$, and subject to the boundary condition (3), we have

$$\begin{array}{cc}\hfill & \mathbf{u}\in L^{\infty }\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right)\cap L^2\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\mathbf{u}}_t\in L^{\infty }\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & p\in L^{\infty }\left((0,\infty );H^1\left(\mathrm{\Omega }\right)\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{lim}{\left|\right|\mathbf{u}\left|\right|}_{L^2}=0.\hfill \end{array}$$

(13)

Theorem 2 is obtained by Lemmas 3–6 and 8.

Remark 2.

Based on Theorem 2, we can see that the large-time asymptotic state of the system (1) in $L^2\left(\mathrm{\Omega }\right)$ space consists only of the hydrostatic balance states.

1.3. Organization of the Paper

The rest of this article is arranged as follows: Section 2 gives linear stability and instability. Section 3 introduces regularity estimates for the Darcy–Boussinesq Equation (7). Section 4 provides the large-time asymptotic behavior for the system (7).

2. Linear Stability and Instability

We consider the linearized equations of the steady-state solution given in the preceding remark, which are given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+{\displaystyle \frac{\mu }{K}}\mathbf{u}=-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ {\displaystyle \frac{\partial \rho }{\partial t}}=-\delta (\mathbf{u}·\nabla )\mathrm{\Psi },\hfill \\ \nabla ·\mathbf{u}=0,\hfill \end{array}\right.\end{array}$$

(14)

which are subject to boundary conditions (3) and initial data (8).

Lemma 2.

Assume that $\delta (x,y)<{\delta }_0<0$ and $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$. For the steady-state solution $(\mathbf{0},{\rho }_s)$ where $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$, it is linearly stable. More precisely, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \underset{t\to \infty }{lim}{\left|\right|\mathbf{u}\left|\right|}_{L^2}^2=0,\underset{t\to \infty }{lim}{\left|\right|{\mathbf{u}}_t\left|\right|}_{L^2}^2=0,\hfill \\ \hfill & \underset{t\to \infty }{lim}{\left|\right|{\rho }_t\left|\right|}_{L^2}^2=0,\underset{t\to \infty }{lim}{\left|\right|\nabla p+\rho \nabla \mathrm{\Psi }\left|\right|}_{L^2}^2=0.\hfill \end{array}\end{array}$$

(15)

Proof. 

Making use of the condition $\delta (x,y)<{\delta }_0<0$, we can rewrite system (14) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+{\displaystyle \frac{\mu }{K}}\mathbf{u}=-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ {\displaystyle \frac{1}{-\delta }}{\displaystyle \frac{\partial \rho }{\partial t}}=(\mathbf{u}·\nabla )\mathrm{\Psi },\hfill \\ \nabla ·\mathbf{u}=0,\hfill \\ {\mathbf{u}·\mathbf{n}|}_{\partial \mathrm{\Omega }}=0.\hfill \end{array}\right.\end{array}$$

(16)

which is subject to boundary conditions (3) and initial data (8).

By multiplying the first and second equations of (44) by $\mathbf{u}$ and $\rho$ in $L^2$ and then adding them together, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}\left({\left|\mathbf{u}\right|}^2+{\displaystyle \frac{{\rho }^2}{-C_a\delta }}\right)dxdy=-{\displaystyle \frac{2\mu }{K{C}_a}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy.\hfill \end{array}\end{array}$$

(17)

Integrating (46) with respect to time gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\left|\right|\mathbf{u}\left|\right|}_{L^2}^2+{{\displaystyle \left|\right|\frac{\rho }{\sqrt{-C_a\delta }}\left|\right|}}_{L^2}^2+{\displaystyle \frac{2\mu }{K{C}_a}}{\int }_0^t{\left|\right|\mathbf{u}\left(t^{\prime }\right)\left|\right|}_{L^2}^{2}d{t}^{\prime }\hfill \\ \hfill & ={\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}^2+{{\displaystyle \left|\right|\frac{{\rho }_0}{\sqrt{-C_a\delta }}\left|\right|}}_{L^2}^2.\hfill \end{array}\end{array}$$

(18)

We then deduce from (17) and (18) that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{t\to \infty }{lim}{\left|\right|\mathbf{u}\left|\right|}_{L^2}^2=0.\end{array}\end{array}$$

(19)

Note that ${\mathbf{u}}_t$ satisfies the follows system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\mathbf{u}}_{tt}+{\displaystyle \frac{\mu }{K}}{\mathbf{u}}_t=-\nabla p_t+\delta \left((\mathbf{u}·\nabla )\mathrm{\Psi }\right)\nabla \mathrm{\Psi },\hfill \\ \nabla ·{\mathbf{u}}_t=0,{\mathbf{u}}_t{·\mathbf{n}|}_{\partial \mathrm{\Omega }}=0.\hfill \end{array}\right.\end{array}$$

(20)

Taking the $L^2$ inner product of (20)₁ with ${\mathbf{u}}_t$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}{\left|{\mathbf{u}}_t\right|}^{2}dxdy& =2{\int }_{\mathrm{\Omega }}\delta \left((\mathbf{u}·\nabla )\mathrm{\Psi }\right)\left(({\mathbf{u}}_t·\nabla )\mathrm{\Psi }\right)\hfill \\ & \le -{\displaystyle \frac{2\mu }{K{C}_a}}{\int }_{\mathrm{\Omega }}{\left|{\mathbf{u}}_t\right|}^{2}dxdy+ϵ{\left|\right|\sqrt{-\delta }\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2{\int }_{\mathrm{\Omega }}{\left|{\mathbf{u}}_t\right|}^{2}dxdy\hfill \\ & +{\displaystyle \frac{{\left|\right|\sqrt{-\delta }\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2}{4ϵ}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy.\hfill \end{array}\end{array}$$

(21)

Choosing $ϵ\le {\displaystyle \frac{\mu }{K{C}_a{\left|\right|\sqrt{-\delta }\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2}}$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}{\left|{\mathbf{u}}_t\right|}^{2}dxdy+{\displaystyle \frac{\mu }{K{C}_a}}{\int }_{\mathrm{\Omega }}{\left|{\mathbf{u}}_t\right|}^{2}dxdy\le {\displaystyle \frac{C_a{\left|\right|\sqrt{-\delta }\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^4}{4\mu }}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy<\infty \end{array}\end{array}$$

(22)

which gives that

$${\left|\right|{\mathbf{u}}_t\left|\right|}_{L^2}^2+{\displaystyle \frac{\mu }{K{C}_a}}{\int }_0^t{\left|\right|{\mathbf{u}}_{t^{\prime }}\left(t^{\prime }\right)\left|\right|}_{L^2}^{2}d{t}^{\prime }={\left|\right|{\mathbf{u}}_t\left(0\right)\left|\right|}_{L^2}^2+{\displaystyle \frac{C_a{\left|\right|\sqrt{-\delta }\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^4}{4\mu }}{\int }_0^t{\left|\right|\mathbf{u}\left(t^{\prime }\right)\left|\right|}_{L^2}^{2}d{t}^{\prime }.$$

We can also deduce from (22) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\left|\right|{\mathbf{u}}_t\left|\right|}_{L^2}^2=0.\end{array}$$

(23)

Finally, based on systems (14) and (19), we obtain the remaining two limits in (15). □

To consider the linear stability of the steady-state solution $(\mathbf{0},{\rho }_s)$ where $\nabla {\rho }_s=\delta (x,y)\nabla \mathrm{\Psi }$, let us consider the following eigenvalue problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\rho }_s\lambda \mathbf{u}=-{\displaystyle \frac{\mu }{K}}\mathbf{u}-\nabla p-\rho \nabla \mathrm{\Psi },\hfill \\ \lambda \rho =-\delta (\mathbf{u}·\nabla )\mathrm{\Psi },\hfill \\ \nabla ·\mathbf{u}=0,\hfill \end{array}\right.\end{array}$$

(24)

from which we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(C_a{\lambda }^2+\lambda {\displaystyle \frac{\mu }{K}}\right)\mathbf{u}=-\lambda \nabla p+\delta (\mathbf{u}·\nabla \mathrm{\Psi })\nabla \mathrm{\Psi },\hfill \\ \nabla ·\mathbf{u}=0,\hfill \\ {\mathbf{u}·\mathbf{n}|}_{\partial \mathrm{\Omega }}=0.\hfill \end{array}\right.\end{array}$$

(25)

To show the existence of instability, we consider the following special cases:

$$\begin{array}{c}\hfill \mathrm{\Psi }=gy,\nabla {\rho }_s={(0,\delta g)}^T{,\mathrm{\Omega }=\mathbb{T}\times [0,h],v|}_{y=0,h}=0.\end{array}$$

(26)

By introducing a stream function, we can eliminate the pressure in (25). To this end, we let

$$\begin{array}{c}\hfill u=-{\partial }_y\psi ,v={\partial }_x\psi ;\end{array}$$

(27)

then, one can infer from (25) that $\psi$ satisfies the follow system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\lambda }^2\mathrm{\Delta }\psi =-\lambda {\displaystyle \frac{\mu }{K}}\mathrm{\Delta }\psi +\delta g^2{\partial }_x^2\psi ,(x,y)\in \mathbb{T}\times (0,h),\hfill \\ {\psi |}_{y=0,h}=0.\hfill \end{array}\right.\end{array}$$

(28)

We can solve the preceding system by setting

$$\begin{array}{c}\hfill \psi =e^{inx}\mathrm{\Psi }\left(y\right).\end{array}$$

(29)

In fact, we can conclude that $\mathrm{\Psi }$ solves the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(C_a{\lambda }^2+\lambda {\displaystyle \frac{\mu }{K}}\right)\left({\displaystyle \frac{d^2}{d{y}^2}}-n^2\right)\mathrm{\Psi }=-\delta g^2{n}^2\mathrm{\Psi },\hfill \\ {\mathrm{\Psi }|}_{y=0,h}=0.\hfill \end{array}\right.\end{array}$$

(30)

It is not hard to see that the ordinary differential Equation (30) only has solutions of the form

$$\begin{array}{c}\hfill \mathrm{\Psi }=Asin{\displaystyle \frac{m\pi }{h}},\end{array}$$

(31)

and $A\ne 0$ if and only if $\lambda$ is the root of

$$\begin{array}{c}\hfill {\lambda }^2+{\displaystyle \frac{\mu }{K{C}_a}}\lambda -{\displaystyle \frac{\delta g^2{n}^2}{C_a\left(\frac{m^2{\pi }^2}{h^2}+n^2\right)}}=0.\end{array}$$

(32)

Hence, $A\ne 0$ if and only if

$$\begin{array}{c}\hfill \lambda ={\lambda }_{mn}^{\pm }=-{\displaystyle \frac{\mu }{2K{C}_a}}\pm \sqrt{{\displaystyle \frac{{\mu }^2}{4K^2{C}_a^2}}+{\displaystyle \frac{\delta g^2{n}^2}{C_a\left(\frac{m^2{\pi }^2}{h^2}+n^2\right)}}}.\end{array}$$

(33)

Based on the exact expressions (33) of the eigenvalues, we have the following proposition.

Proposition 1.

Assume that $\mathrm{\Psi }=gy$. For the steady-state solution $(\mathbf{0},{\rho }_s)$ satisfying ${\rho }_s={(0,{\delta }_{0}g)}^T$, it is linearly unstable if ${\delta }_0>0$.

3. Regularity Estimates

All $L^q$-norms of $\rho$ are conserved if $\mathbf{u}$ is sufficiently smooth, i.e., for any $1\le q\le \infty$,

$$\begin{array}{c}\hfill {\left|\right|\rho +{\rho }_s\left|\right|}_{L^q}={\left|\right|{\rho }_0+{\rho }_s\left|\right|}_{L^q}\end{array}$$

(34)

for all $t\ge 0$ if ${\rho }_0\in L^q\left(\mathrm{\Omega }\right)$. In fact, for the case of $q\in 2\mathbb{N}$, one can infer from Equation (7) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d{\left|\right|\rho +{\rho }_s\left|\right|}_{L^q}^q}{dt}}& ={\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}{\left|\rho +{\rho }_s\right|}^{q}dxdy=-q{\int }_{\mathrm{\Omega }}{(\rho +{\rho }_s)}^{q-1}\mathbf{u}·\nabla (\rho +{\rho }_s)dxdy\hfill \\ & =-{\int }_{\mathrm{\Omega }}\mathbf{u}·\nabla {(\rho +{\rho }_s)}^{q}dxdy\hfill \\ & =-{\int }_{\mathrm{\Omega }}{(\rho +{\rho }_s)}^q\mathbf{u}·\mathbf{n}dxdy+{\int }_{\mathrm{\Omega }}{(\rho +{\rho }_s)}^q\nabla ·\mathbf{u}dxdy=0\hfill \end{array}\end{array}$$

(35)

where we have used $\nabla ·\mathbf{u}=0$ and boundary condition (3). Using the fact that the $L^2$-norm of $\rho$ is conserved, one can directly prove that the bounds for the energy of the fluid with governing equations are those in Equation (1), which are given in the following lemma.

3.1. Estimates of ${\left|\right|\mathbf{u}\left|\right|}_{L^{\infty }\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right)}$

Lemma 3.

For the solution $(\mathbf{u},\rho )$ of Equation (7) subject to boundary condition (3) and with the initial condition $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$, $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$, we then have

$$\begin{array}{c}\hfill \mathbf{u}\in L^{\infty }\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right)\end{array}$$

(36)

and it also holds that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left|\right|\mathbf{u}\left|\right|}_{L^2}^2\le e^{-{\displaystyle \frac{\mu }{K{C}_a}}t}{\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}^2+{\displaystyle \frac{K{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2}{{\mu }^2}}{\left|\right|{\rho }_0\left|\right|}_{L^2}^2.\end{array}\end{array}$$

(37)

Proof. 

By multiplying the first equation of (1) by $\mathbf{u}$, after integration by parts, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & C_a{\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{{\left|\mathbf{u}\right|}^2}{2}}dxdy+{\displaystyle \frac{\mu }{K}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy\hfill \\ & =-{\int }_{\mathrm{\Omega }}\rho \nabla \mathrm{\Psi }·\mathbf{u}dxdy\hfill \\ \hfill & \le ϵ{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}{\mathbf{u}}_{L^2}^2+{\displaystyle \frac{{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}}{4ϵ}}{\left|\right|\rho \left|\right|}_{L^2}^2.\hfill \end{array}\end{array}$$

(38)

By choosing $ϵ\le {\displaystyle \frac{\mu }{2{\left|\right|\nabla f\left|\right|}_{L^{\infty }}}}$, we then have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy+{\displaystyle \frac{\mu }{2K{C}_a}}{\left|\right|\mathbf{u}\left|\right|}_{L^2}^2\le {\displaystyle \frac{{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2}{2\mu C_a}}{\left|\right|{\rho }_0\left|\right|}_{L^2}^2.\hfill \end{array}\end{array}$$

(39)

Then, by applying Gronwall’s inequality, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left|\right|\mathbf{u}\left|\right|}_{L^2}^2\le e^{-{\displaystyle \frac{\mu }{K{C}_a}}t}{\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}^2+{\displaystyle \frac{K{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}^2}{{\mu }^2}}{\left|\right|{\rho }_0\left|\right|}_{L^2}^2.\end{array}\end{array}$$

(40)

□

3.2. Estimates of ${\left|\right|\mathbf{u}\left|\right|}_{L^2\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right)}$

Lemma 4.

For the solution $(\mathbf{u},\rho )$ of Equation (7) with initial data $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$, $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$, we then have

$$\begin{array}{c}\hfill \mathbf{u}\in L^2\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right).\end{array}$$

(41)

Proof. 

Let us introduce the following new variables:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathbf{v}=\mathbf{u},\hfill \\ \rho =\theta +e(x,y),\hfill \\ p=q+h(x,y),\hfill \end{array}\right.\end{array}$$

(42)

where $e(x,y)$ and $h(x,y)$ are given by the following equations:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & e(x,y)=-\gamma \mathrm{\Psi }(x,y)+\beta ,\gamma >0,\hfill \\ \hfill & \nabla h=-e(x,y)\nabla \mathrm{\Psi }.\hfill \end{array}\end{array}$$

(43)

Then, one can see that $(\mathbf{v},\theta ,q)$ solve the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_a{\displaystyle \frac{\partial \mathbf{v}}{\partial t}}+{\displaystyle \frac{\mu }{K}}\mathbf{v}=-\nabla p-\theta \nabla \mathrm{\Psi },\hfill \\ {\displaystyle \frac{\partial \theta }{\partial t}}+(\mathbf{v}·\nabla )\theta =\gamma (\mathbf{v}·\nabla )\mathrm{\Psi },\hfill \\ \nabla ·\mathbf{v}=0,\hfill \end{array}\right.\end{array}$$

(44)

which are subject to the following boundary conditions:

$$\begin{array}{cc}\hfill & {\mathbf{v}·\mathbf{n}|}_{\partial \mathrm{\Omega }}=0.\hfill \end{array}$$

(45)

By multiplying the first and second equations of (44) by $\gamma \mathbf{v}$ and $\theta$ and then adding them together, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}\left({\gamma \left|\mathbf{v}\right|}^2+{\theta }^2\right)dxdy=-{\displaystyle \frac{2\gamma \mu }{K{C}_a}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{v}\right|}^{2}dxdy\hfill \end{array}\end{array}$$

(46)

Integrating (46) with respect to time gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \gamma {\left|\right|\mathbf{v}\left|\right|}_{L^2}^2+{\left|\right|\theta \left|\right|}_{L^2}^2+2{\displaystyle \frac{\gamma \mu }{K{C}_a}}{\int }_0^t{\left|\right|\mathbf{v}\left(t^{\prime }\right)\left|\right|}_{L^2}^{2}d{t}^{\prime }\hfill \\ \hfill & =\gamma {\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}^2+{\left|\right|{\theta }_0\left|\right|}_{L^2}^2,{\theta }_0={\rho }_0+\gamma \mathrm{\Psi }-\beta ,\hfill \end{array}\end{array}$$

(47)

from which we can obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_0^t{\left|\right|\mathbf{v}\left(t^{\prime }\right)\left|\right|}_{L^2}^{2}d{t}^{\prime }\le {\displaystyle \frac{K{C}_a}{2\gamma \mu }}\left(\gamma {\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}^2+{\left|\right|{\theta }_0\left|\right|}_{L^2}^2\right).\end{array}\end{array}$$

(48)

Because the right-hand side of (48) is independent of t, we therefore obtain

$$\begin{array}{c}\hfill \mathbf{u}\in L^2\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right).\end{array}$$

□

3.3. Estimates of ${\left|\right|p\left|\right|}_{L^{\infty }\left((0,\infty );H^1\left(\mathrm{\Omega }\right)\right)}$

Lemma 5.

For the solution $(\mathbf{u},\rho )$ of Equation (7) subject to boundary condition (3) and with the initial data $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$, $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$, we then have

$$\begin{array}{c}\hfill p\in L^{\infty }\left((0,\infty );H^1\left(\mathrm{\Omega }\right)\right).\end{array}$$

(49)

Proof. 

Note that the pressure solves the following equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{\Delta }p=-\nabla ·(\rho \nabla \mathrm{\Psi }),\hfill \\ \nabla p·{\mathbf{n}}_{\partial \mathrm{\Omega }}=-\rho \nabla \mathrm{\Psi }·{\mathbf{n}}_{\partial \mathrm{\Omega }}.\hfill \end{array}\right.\end{array}$$

(50)

By multiplying the first equation of (50) by p and carrying out an integration by parts, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{\mathrm{\Omega }}p\mathrm{\Delta }pdxdz& ={\int }_{\mathrm{\Omega }}\rho \nabla p·\nabla \mathrm{\Psi }dxdz-{\int }_{\partial \mathrm{\Omega }}p\rho \nabla \mathrm{\Psi }·\mathbf{n}ds,\hfill \end{array}\end{array}$$

(51)

where we have used $\mathrm{\Delta }\mathrm{\Psi }=0$ and

$${\int }_{\partial \mathrm{\Omega }}p\rho \nabla \mathrm{\Psi }·\mathbf{n}ds={\int }_{\mathrm{\Omega }}p\nabla \rho ·\nabla \mathrm{\Psi }dxdz+{\int }_{\mathrm{\Omega }}p\rho \mathrm{\Delta }\mathrm{\Psi }dxdz+{\int }_{\mathrm{\Omega }}\rho \nabla p·\nabla \mathrm{\Psi }dxdz.$$

Note that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{\mathrm{\Omega }}p\mathrm{\Delta }pdxdz& =-{\int }_{\mathrm{\Omega }}{\left|\nabla p\right|}^{2}dxdz+{\int }_{\partial \mathrm{\Omega }}p\mathbf{n}·\nabla pdx.\hfill \end{array}\end{array}$$

(52)

Combining (51) and (52), we find that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{\mathrm{\Omega }}{\left|\nabla p\right|}^{2}dxdz& =-{\int }_{\mathrm{\Omega }}\rho \nabla p·\nabla \mathrm{\Psi }dxdz\hfill \\ \hfill & \le {\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}{\left|\right|p\left|\right|}_{H^1}{\left|\right|\rho \left|\right|}_{L^2}\hfill \\ & \le ϵ{\left|\right|p\left|\right|}_{H^1}^2+C_{ϵ}{\left|\right|\rho \left|\right|}_{L^2}^2.\hfill \end{array}\end{array}$$

(53)

As p is only defined up to a constant, we choose it such that it is average-free, and combining this, we finally establish the following inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left|\right|p\left|\right|}_{H^1}^2\le C_{ϵ}{\left|\right|\rho \left|\right|}_{L^2}^2<\infty .\end{array}\end{array}$$

(54)

□

3.4. Estimates of ${\left|\right|{\mathbf{u}}_t\left|\right|}_{L^2\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right)}$ and ${\left|\right|{\mathbf{u}}_t\left|\right|}_{L^{\infty }\left((0,\infty ;L^2\left(\mathrm{\Omega }\right)\right)}$

Lemma 6.

For the solution $(\mathbf{u},\rho )$ of Equation (7) subject to boundary condition (3) and with the initial condition $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$, $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$, we then have

$$\begin{array}{c}\hfill {\mathbf{u}}_t\in L^{\infty }\left((0,\infty );L^2\left(\mathrm{\Omega }\right)\right).\end{array}$$

(55)

Proof. 

Making use of the first equation of (1), we have

$${\left|\right|{\mathbf{u}}_t\left|\right|}_{L^2}\le {\displaystyle \frac{\mu }{K}}{\left|\right|\mathbf{u}\left|\right|}_{L^2}+{\left|\right|\nabla p\left|\right|}_{L^2}+{\left|\right|\nabla \mathrm{\Psi }\left|\right|}_{L^{\infty }}{\left|\right|\nabla p\left|\right|}_{L^2}\le C\left({\left|\right|{\rho }_0\left|\right|}_{L^2}+{\left|\right|{\mathbf{u}}_0\left|\right|}_{L^2}\right).$$

□

4. Large-Time Behavior

Lemma 7

([31]). Assume $g\left(t\right)\in L^1(0,\infty )$ is a non-negative and uniformly continuous function. Then,

$$g\left(t\right)\to 0\mathrm{as}t\to \infty .$$

In particular, if $g\left(t\right)\in L^1(0,\infty )$ is non-negative and satisfies this condition for any $0\le s<t<\infty$, such that $\left|g\left(t\right)-g\left(s\right)\right|\le C(t-s)$ for some constant C, then

$$g\left(t\right)\to 0\mathrm{as}t\to \infty .$$

Lemma 8.

For the solution $(\mathbf{u},\rho )$ of Equation (1) with the initial condition $({\mathbf{u}}_0,{\rho }_0)\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$, $\nabla \mathrm{\Psi }\in L^{\infty }\left(\mathrm{\Omega }\right)$, we then have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\left|\right|\mathbf{u}\left|\right|}_{L^2}=0.\end{array}$$

(56)

Proof. 

We let $g\left(t\right)={\left|\right|\mathbf{u}\left|\right|}_{L^2}^2$. We can then infer from Lemma 4 that

$$g\left(t\right)\in L^1(0,\infty ).$$

Based on

$$\begin{array}{cc}& {\displaystyle \frac{d}{dt}}{\int }_{\mathrm{\Omega }}\left({\gamma \left|\mathbf{u}\right|}^2+{\theta }^2\right)dxdy=-{\displaystyle \frac{2\gamma \mu }{K{C}_a}}{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^{2}dxdy\in L^{\infty }\left(0,\infty \right),\hfill \end{array}$$

one can see that there exists a constant C such that $\left|g\left(t\right)-g\left(s\right)\right|\le C(t-s)$. Thus, Lemma 7 gives an asymptotic result (56). One can also obtain this result by applying Gronwall’s inequality. □

5. Discussion

Previous studies have mainly analyzed the linear stability and nonlinear perturbation decay rate of linear steady-state solutions under constant gravity. Moreover, the steady-state solutions considered in previous studies are top-light and bottom-heavy.

In this study, the following developments were made on top of previous studies. First, the linear stability and instability of the general steady-state solution under variable gravity were considered. Second, the asymptotic properties of the disturbance of the general steady-state solution were analyzed. The results show that for an unstable steady-state solution, the disturbance speed will also approach zero.

Our findings could have some applications in geophysics and engineering. For example, in the field of subsurface contaminant transport, our results suggest that engineered barriers with controlled permeability gradients $(\delta <0)$ could stabilize plume migration, while unintended $\delta >0$ regions (e.g., near faults) risk runaway instability, and in the field of CO₂ sequestration, ensuring $\delta <0$ in saline aquifers could prevent caprock failure due to buoyancy-driven instabilities.

While our analysis rigorously addresses linear stability and asymptotic convergence, there are also some other research directions worth expanding on, such as proving that the asymptotic state must be top-light and bottom-heavy and determining the mathematical expression of the asymptotic state.

6. Conclusions

This study advances the mathematical foundations of porous media flows, revealing that inertia plays a dual role in stabilizing or destabilizing buoyancy-driven systems. By resolving paradoxes between transient growth and large-time convergence, our work provides a robust framework for predicting and controlling fluid behavior in diverse applications, from energy extraction to environmental protection. Future efforts should focus on bridging the gap between linear theory and nonlinear reality, with an eye toward multiscale, multiphysics simulations of complex geological systems.