Entropy Geometry and Augmented Mobility for Reactive Maxwell--Stefan Membrane Transport with Finite Occupancy

Technical Summary: Entropy Geometry and Augmented Mobility for Reactive Maxwell–Stefan Membrane Transport with Finite Occupancy

Problem Statement
This paper investigates a reactive Maxwell–Stefan-type membrane transport system constrained by finite occupancy. Unlike classical dilute mixture models where the state space is an unbounded simplex, the relevant state space here is a bounded domain $D = \{u \in (0, 1)^n : \sum u_i < 1\}$. The quantity $1 - \varrho(u)$ represents the local vacancy fraction (free volume), which becomes critical as the system approaches saturation.

The primary analytical challenge arises from the thermodynamic structure induced by this bounded geometry. The natural entropy is of the Boltzmann–Fermi type, leading to entropy variables $w_i = \log u_i - \log(1 - \varrho(u))$. In this framework, the species space naturally splits into two channels:

1. Composition Channel: Redistribution of species at a fixed total occupancy.
2. Mass Channel: Collective variation of the total filling fraction (vacancy change).

The authors identify a structural defect in standard Maxwell–Stefan models under these constraints: the mobility matrix may be coercive on the composition subspace but fail to control the mass channel (the direction of total occupancy variation). This loss of coercivity prevents the closure of entropy estimates required for proving global existence and stability.

Methodology
The paper employs an entropy-based framework, utilizing the convexity of the Boltzmann–Fermi entropy to parametrize the state space via unconstrained entropy variables. The core methodological innovation is the introduction of a rank-one augmentation to the mobility matrix.

1. Augmented Mobility: The authors propose modifying the projected mobility $B(u)$ by adding a term $\gamma_0 \mathbf{1} \otimes \mathbf{1}$ (where $\gamma_0 > 0$). This augmentation acts exclusively in the mass channel, restoring full coercivity without altering the composition block.
2. Continuum Analysis:
   • Existence: Global entropy weak solutions are constructed using an implicit Rothe scheme formulated in entropy variables. This choice ensures the discrete solutions remain within the admissible state space $D$ at every time step.
   • Stability: A weak–strong stability estimate is derived using a relative entropy argument. The augmented mobility provides the necessary dissipation to control the difference between a weak solution and a strong solution.
3. Discrete Analysis:
   • A fully implicit finite-volume scheme is developed in entropy variables.
   • The scheme is designed to preserve the bounded-occupancy structure ($u_K \in D$) and satisfy a discrete entropy inequality.
   • Convergence of the discrete solutions to the entropy weak solution is proven as the mesh and time step are refined.

Key Contributions and Results
The paper establishes four main results:

1. Quantitative Channel-wise Coercivity: The authors prove that the rank-one augmentation $\gamma_0 \mathbf{1} \otimes \mathbf{1}$ restores full coercivity to the mobility matrix. Specifically, the augmented mobility satisfies a lower bound proportional to the squared norm of the gradient of entropy variables, split into composition and mass components.
2. Global Existence of Weak Solutions: Using the coercivity estimate and the implicit Rothe scheme, the paper proves the existence of global entropy weak solutions for the reactive system with nonlinear sink terms and linear conversion.
3. Weak–Strong Stability and Uniqueness: A relative entropy estimate is derived, showing that the distance between a weak solution and a strong solution (staying in a compact subset of $D$) grows at most exponentially. This implies uniqueness within the class of strong solutions.
4. Convergence of Finite-Volume Approximation: The paper demonstrates that a fully implicit finite-volume scheme, which respects the entropy structure and state constraints, converges to the entropy weak solution. The discrete scheme satisfies a discrete entropy inequality analogous to the continuous case.

Significance and Claims
The paper positions itself as a rigorous continuum-and-discrete entropy theory for a specific class of finite-capacity reactive membrane transport models. The authors explicitly state that the model is a "prototype" intended to isolate the minimal structural core required for analytical stability under bounded occupancy, explicit vacancies, and reactive conversion.

The significance of the work lies in identifying that a single, minimal structural modification (the rank-one augmentation) is sufficient to repair the loss of coercivity inherent in finite-occupancy Maxwell–Stefan systems. This repair allows for the application of standard entropy methods to prove existence, stability, and numerical convergence, bridging the gap between the thermodynamic geometry of bounded domains and the analytical requirements of cross-diffusion systems. The authors do not claim to provide a complete constitutive derivation for all specific membrane friction mechanisms but rather to establish a mathematically stable framework for the structural core of such problems.