Asymptotic stability of Stationary solutions to 3D incompressible flow in porous media with diffusion

Technical Summary: Asymptotic Stability of Stationary Solutions to 3D Incompressible Flow in Porous Media with Diffusion

Problem Statement
This paper investigates the asymptotic behavior of the Diffusive Porous Media Equation (DPME) in $\mathbb{R}^3$. The governing system describes an incompressible flow diffusing through a porous medium, modeled by the temperature field $\theta$, velocity $V_\theta$, and an external force $f$:
$$\begin{cases} \partial_t\theta + V_\theta \cdot \nabla\theta = \Delta\theta + f, \\ V_\theta = -(\nabla P_\theta + \theta e_3), \\ \nabla \cdot V_\theta = 0, \end{cases}$$
where $e_3 = (0,0,1)$ represents the direction of gravity. The study focuses on two primary objectives: (1) establishing the existence and uniqueness of stationary solutions ($\theta_{st}$) to the system under specific constraints on the external force $f$, and (2) proving the asymptotic stability of these stationary solutions, specifically estimating the decay rate of perturbations $\omega = \theta - \theta_{st}$ as $t \to \infty$.

Methodology
The paper employs a combination of functional analysis, iterative construction techniques, and harmonic analysis methods:

1. Existence of Stationary Solutions:

   • The authors adapt the approach developed by Bjorland and Schonbek for the Navier-Stokes equations.
   • They construct a sequence of approximate solutions by iteratively solving linear parabolic PDEs and associated linear elliptic equations.
   • Specifically, they define a linear parabolic equation (3.1) and a linear stationary equation (3.2). By integrating the solution of the parabolic equation in time, they relate it to the solution of the stationary equation.
   • Uniqueness is established within a specific ball of the homogeneous Sobolev space $\dot{H}^1(\mathbb{R}^3)$, relying on the smallness of the external force $f$ in the norm $\|f\|_X = \max\{\|f\|_{L^2}, \|f\|_{\dot{H}^{-1}}\}$.

2. Decay Character and Fourier Splitting:

   • The concept of "decay character" ($r^*$) is central to the analysis. Defined by Bjorland and Schonbek, $r^*$ quantifies the algebraic order of the Fourier transform of a function near the origin ($|\hat{g}(\xi)| \sim |\xi|^{r^*}$ as $\xi \to 0$).
   • The Fourier Splitting Method (developed by Schonbek) is utilized to derive differential inequalities for the $L^2$ norm of the perturbation $\omega$. This involves splitting the frequency space into a time-dependent ball $B(R(t))$ and its complement to isolate the influence of low frequencies on the long-time decay.

3. Existence of Perturbation Solutions:

   • To handle the nonlinear terms in the perturbation equation (1.3), the authors use the retarded mollifiers method (Caffarelli, Kohn, and Nirenberg). This technique regularizes the convective terms by introducing a time-delayed mollifier, allowing for the construction of weak solutions via an iterative process on time strips, followed by a convergence argument using energy estimates and the Aubin-Lions lemma.

Key Contributions and Results

• Theorem 1.1 (Existence and Uniqueness of Stationary Solutions):
  The paper proves that if the external force $f \in L^2(\mathbb{R}^3) \cap \dot{H}^{-1}(\mathbb{R}^3)$ has a decay character $\gamma^* > 2$ and satisfies a smallness condition $\|f\|_X \leq C_M$, there exists a unique weak stationary solution $\theta_{st} \in \dot{H}^1(\mathbb{R}^3)$. The solution satisfies the bound $\|\theta_{st}\|_{L^2} \leq M$.

• Theorem 1.2 (Asymptotic Stability and Decay Rates):
  The main result establishes that the stationary solution $\theta_{st}$ is asymptotically stable. For any weak solution $\theta$ to the time-dependent DPME with initial data $\theta_0$, the perturbation $\omega = \theta - \theta_{st}$ decays in the $L^2$ norm according to:
  $$\|\omega(t)\|_{L^2}^2 \leq C(1+t)^{-\min\left\{\frac{3}{2}, \frac{3}{2} + \min\{r^*(\theta_0), r^*(\theta_{st})\}\right\}}$$
  Crucially, the initial perturbation $\omega_0$ does not need to be small; only the external force $f$ must be sufficiently small to ensure the existence of the stationary state. The decay rate is determined by the decay characters of both the initial data and the stationary solution.

• Technical Lemmas:
  The paper provides detailed proofs regarding the decay properties of auxiliary linear equations (Theorem 3.5) and the relationship between the time-integrated solution of the parabolic equation and the stationary solution (Lemma 3.7), which are essential for the iterative convergence proof.

Significance and Claims
The paper claims to extend the understanding of porous media flows by rigorously addressing the stability of stationary states in the presence of diffusion and external forcing. By utilizing the decay character concept, the authors provide a precise characterization of how the low-frequency behavior of the initial data and the forcing term dictates the long-time asymptotic decay of the system.

The work relies on established mathematical frameworks (Bjorland-Schonbek for existence, Schonbek for decay, Caffarelli-Kohn-Nirenberg for regularization) but applies them specifically to the 3D Diffusive Porous Media Equation to derive optimal decay rates. The authors do not propose new physical applications or experimental validations; rather, the significance lies in the mathematical proof of well-posedness and the quantitative description of the asymptotic stability of the system under the hypothesis of finite $L^2$ energy and specific low-frequency behavior of the external force.