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A domain hemivariational inequality for 2D and 3D convective Brinkman-Forchheimer extended Darcy equations

This paper establishes the existence, energy equality, and uniqueness of weak solutions to non-stationary 2D and 3D convective Brinkman-Forchheimer extended Darcy equations formulated as domain hemivariational inequalities, utilizing a regularized Galerkin approximation scheme that also extends to the three-dimensional Navier-Stokes equations.

Original authors: Jyoti Jindal, Sagar Gautam, Manil T. Mohan

Published 2026-03-31

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CC0 1.0

Original authors: Jyoti Jindal, Sagar Gautam, Manil T. Mohan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a thick, sticky fluid (like honey mixed with sand) flows through a sponge. This isn't just a simple flow; it's a chaotic dance where the fluid pushes itself, gets slowed down by the sponge's holes, and sometimes even gets "pumped" or "sucked" by internal forces.

This paper is a mathematical detective story about solving the equations that describe this chaotic flow, but with a twist: the rules of the game aren't perfectly smooth. Sometimes, the forces acting on the fluid change abruptly, like a light switch flipping on and off, rather than a dimmer switch sliding smoothly.

Here is a breakdown of what the authors did, using everyday analogies:

1. The Setting: The "Sponge" and the "Fluid"

The authors are studying the CBFeD equations. Think of this as a super-charged version of the famous Navier-Stokes equations (which describe how water or air moves).

The Fluid: It's moving through a porous medium (like a sponge or soil).
The Damping: The sponge resists the flow. The faster the fluid moves, the harder the sponge pushes back. This is like running through deep water; the faster you go, the more resistance you feel.
The Pumping: The authors added a "pumping" term. Imagine if, in certain spots, the sponge actually tried to push the fluid along, counteracting the resistance.
The Twist (Hemivariational Inequality): Usually, math problems assume forces change smoothly. But in the real world, things can be "jagged." Maybe the friction changes suddenly when the fluid hits a specific speed. The authors call this a Hemivariational Inequality. It's like trying to predict the path of a ball rolling on a surface that has sudden, jagged bumps and cliffs, rather than a smooth hill.

2. The Problem: "Jagged" Rules Make Math Hard

In standard physics, if you know the starting position and the smooth rules, you can usually predict the future. But when the rules are "jagged" (non-smooth) and happen everywhere inside the sponge (not just on the edges), the math gets incredibly messy.

The Challenge: The authors had to prove two things:
1. Existence: Does a solution even exist? (i.e., Is there any way the fluid can move that satisfies these jagged rules?)
2. Uniqueness: Is there only one way the fluid can move? (i.e., If we start with the same conditions, will the fluid always end up in the same place, or could it split into two different realities?)

3. The Solution: The "Pixelated" Approach (Galerkin Method)

To solve this, the authors used a technique called the Galerkin method.

The Analogy: Imagine trying to draw a perfect circle on a computer screen. You can't do it perfectly with pixels. Instead, you start with a few big pixels (a rough square), then more pixels (an octagon), then even more (a 16-sided shape), and so on. As you add more pixels, the shape gets closer and closer to a perfect circle.
In the Paper: The authors broke the complex, infinite-dimensional fluid problem down into smaller, simpler "pixelated" versions (finite-dimensional problems). They solved these simple versions first, proved they behaved well, and then showed that as they added more "pixels" (making the approximation finer and finer), the solution settled down into a real, valid answer for the full, complex problem.

4. The Big Wins

The authors achieved several significant breakthroughs:

No "Magic" Restrictions: Previous studies often required the fluid to be very "thick" (high viscosity) or the sponge to be very specific to prove a solution existed. This paper proved solutions exist regardless of how thick the fluid is or how strong the sponge's resistance is. They removed the "magic numbers" that previous mathematicians needed.
Energy Equality: In physics, energy is usually conserved (or lost to heat in a predictable way). The authors proved that even with these jagged, non-smooth rules, the fluid still obeys the law of energy conservation. This is like proving that even if you drive a car over a bumpy, jagged road, your fuel gauge still drops in a predictable, calculable way.
Uniqueness: They proved that for most realistic scenarios (and even some tricky ones), there is only one possible outcome. If you set up the experiment twice, you get the exact same result both times. This is crucial for engineers who need to trust their models.

5. Why Does This Matter?

This isn't just abstract math. These equations model real-world problems:

Oil and Gas: How oil flows through underground rock formations.
Groundwater: How water moves through soil and aquifers.
Blood Flow: How blood moves through porous tissues or artificial filters.

By proving that these complex, "jagged" models have reliable, unique solutions, the authors gave engineers and scientists a solid mathematical foundation. They can now trust that their computer simulations of fluid flow through porous media are not just guesses, but mathematically sound predictions.

In a nutshell: The authors took a messy, jagged, and difficult math problem about fluid flow through sponges, smoothed out the rough edges using a clever "pixel-by-pixel" strategy, and proved that the universe behaves in a predictable, unique, and energy-conserving way, even when the rules seem to jump around.

1. Problem Statement

The paper investigates the mathematical analysis of non-stationary (time-dependent) Convective Brinkman-Forchheimer Extended Darcy (CBFeD) equations in bounded domains ( $D \subset \mathbb{R}^d$ for $d=2,3$ ). These equations model the flow of viscous incompressible fluids through saturated porous media, incorporating both damping and pumping mechanisms.

The governing system is given by:
$\begin{cases} \partial_t y - \mu \Delta y + (y \cdot \nabla)y + \alpha|y|^{q-1}y + \beta|y|^{r-1}y + \nabla \pi = f + g, \\ \nabla \cdot y = 0, \\ y = 0 \text{ on } \partial D, \quad y(0) = y_0, \end{cases}$
where:

$y$ is the velocity field, $\pi$ is pressure, and $f$ is a given external force.
$\mu > 0$ is the effective viscosity (Brinkman coefficient).
$\beta > 0$ is the Forchheimer coefficient (damping term).
$\alpha < 0$ represents a pumping mechanism (counteracting damping).
$r \ge 1$ is the absorption exponent (with $r=3$ being critical).
$q \in [1, r)$ governs the pumping strength.

The Novelty of the Model:
The forcing term $g$ is not a standard function but satisfies a domain hemivariational inequality (HVI):
$-g(x,t) \in \partial_C j(x, t, y(x,t))$
where $\partial_C j$ is the Clarke subdifferential of a locally Lipschitz superpotential $j$ . Unlike boundary HVIs (where non-smoothness occurs only at the boundary), this domain HVI implies that non-smooth and non-convex effects influence the system throughout the entire interior of the spatial domain $D$ .

2. Methodology

The authors employ a rigorous analytical framework combining functional analysis, non-smooth calculus, and approximation schemes.

Functional Framework: The problem is set in a Gelfand triple involving Sobolev spaces $V$ (divergence-free $H^1_0$ ), $H$ (divergence-free $L^2$ ), and $L^p$ spaces. The authors define linear, bilinear, and nonlinear operators corresponding to the Laplacian, convection, and absorption/pumping terms.
Clarke Subdifferential Regularization: Since the Clarke subdifferential is multivalued and non-smooth, the authors cannot apply standard Galerkin methods directly. They introduce a regularization technique:
1. Approximate the discontinuous/non-smooth function $\theta$ (related to the subdifferential) using a mollifier to create a smooth, single-valued function $\theta_\varepsilon$ .
2. Construct a family of regularized finite-dimensional problems.
Galerkin Approximation:
- The infinite-dimensional problem is projected onto finite-dimensional subspaces spanned by the first $n$ eigenfunctions of the Stokes operator.
- Existence of Approximate Solutions: Proven using Carathéodory's existence theorem for ODEs.
- A Priori Estimates: Uniform energy estimates are derived for the approximate solutions. A key difficulty addressed is handling the integral term involving the regularized subdifferential. The authors use a specific lemma (Lemma 3.6) to bound this term without imposing restrictive conditions on the viscosity $\mu$ .
Passage to the Limit:
- Using the Banach-Alaoglu theorem, weakly convergent subsequences are extracted.
- The Aubin-Lions compactness lemma is used to obtain strong convergence in $L^2(0,T; H)$ , which is crucial for handling the nonlinear convection term $(y \cdot \nabla)y$ .
- The limit is shown to satisfy the original hemivariational inequality by utilizing properties of the Clarke subdifferential and Egorov's theorem.

3. Key Contributions and Results

A. Existence of Weak Solutions

The paper establishes the existence of weak solutions for the non-stationary 2D and 3D CBFeD equations with domain HVIs for all admissible absorption exponents $r \ge 1$ and effective viscosity $\mu > 0$ .

Significance: Unlike previous works (e.g., Migórski et al.) that required restrictive conditions on the viscosity parameter (e.g., $\mu > C \|l\|$ ) to handle the HVI, this work proves existence without any such restriction on $\mu$ .

B. Energy Equality

The authors prove that the weak solutions satisfy the energy equality (conservation of energy in the integral sense) under specific conditions:

2D: For all $r \ge 1$ .
3D: For $r \ge 3$ .
Method: They utilize an approximation technique involving finite-dimensional spaces spanned by the first $n$ eigenfunctions of the Stokes operator (following the approach in [12]), rather than relying on the standard Galerkin approximation which often fails to preserve energy equality in 3D for certain exponents.
Note: For $r \in [1, 3)$ in 3D, the energy equality is not established due to the lack of sufficient regularity in the nonlinear terms.

C. Uniqueness of Solutions

The paper proves the uniqueness of weak solutions, a significant advancement as uniqueness for 3D Navier-Stokes type problems is generally an open problem.

2D: Unique for all $r \ge 1$ .
3D: Unique for $r \ge 3$ .
Critical Case ( $r=3$ in 3D): Uniqueness holds provided the additional condition $2\beta\mu > 1$ is met.
Mechanism: The proof relies on the dissipative nature of the absorption term $\beta|y|^{r-1}y$ and a monotonicity condition on the subdifferential function (Hypothesis 3.4 and condition 3.63). The authors show that the absorption term provides enough control to overcome the nonlinearity of the convection term, unlike in classical NSE where uniqueness is unknown for 3D.

4. Significance and Impact

Bridging a Theoretical Gap: This is one of the first studies to address domain-type hemivariational inequalities for CBFeD equations. Previous literature largely focused on boundary-type HVIs or standard NSE.
Relaxing Viscosity Constraints: By avoiding artificial restrictions on the viscosity coefficient $\mu$ , the results are more physically applicable to a wider range of porous media flows.
Extension to Navier-Stokes: The analysis extends to the 3D non-stationary Navier-Stokes equations (the case where $\alpha=\beta=0$ is not directly covered, but the framework with damping $\beta > 0$ provides a pathway to understanding NSE with damping). Specifically, the results for $r \ge 3$ in 3D offer a class of modified NSE where global existence and uniqueness are rigorously proven.
Mathematical Rigor: The paper provides a robust template for handling non-smooth, non-convex terms in fluid dynamics using Clarke subdifferentials and regularization, offering a blueprint for future studies on complex fluid-structure interactions and contact problems in porous media.

5. Conclusion

The paper successfully establishes the well-posedness (existence, energy equality, and uniqueness) of a complex fluid flow model governed by CBFeD equations with non-smooth domain forces. The work advances the mathematical theory of hemivariational inequalities in fluid mechanics, demonstrating that the inclusion of damping and pumping mechanisms allows for strong analytical results even in three dimensions, where classical Navier-Stokes problems remain intractable regarding uniqueness.