Thermodynamically Consistent Modeling and Stable ALE Approximations of Reactive Semi-Permeable Interfaces

The paper’s Theorem 2.1 states exactly the two mass conservations and the free-energy dissipation identity in the nondimensional model, including the same Δ-terms (Δk, ΔC, ΔK, Δa, Δc, Δm), all shown to be nonnegative by the same constitutive choices and detailed-balance/monotonicity arguments. The derivation proceeds by differentiating the kinetic, bulk, and interfacial energies, using the Laplace–Young jump, surface/bulk transport theorems, and closed-interface geometry to cancel capillary work with surface terms, precisely as in the candidate solution’s outline. The model’s mass invariants ms(AΓ,BΓ,0,0) and ms(AΓ,0,CΓ,C) and the energy identity dEtot/dt = −(Δk+ΔC+ΔK+Δa+Δc+Δm) are explicitly stated and proved in the paper (mass and energy: Theorem 2.1; system and scalings in Eqs. (2.27)–(2.31)) . The energy derivation (Eqs. (2.15)–(2.18)) and the sign conditions via Lemmas B.1–B.3 are the same ingredients used by the model solution . Boundary terms vanish under the stated boundary conditions (u=0 on ∂Ω1, T·n=0 on ∂Ω2, n·∇C+=0), matching the model solution’s assumptions . Hence both are correct and follow substantially the same proof strategy.