Immersed Boundary Electrodiffusion in the Wasserstein Gradient Flow Formalism: Global Well-Posedness, Non-Equilibrium Thermodynamics, and Singular Asymptotics
Abstract
We introduce a unified optimal transport framework for the Immersed Boundary (IB) electrodiffusion active pump model. By representing membrane chemical potential barriers and active pump potentials via smoothed Eulerian kernels (Ψw, ℋw), we formulate the passive dynamics as an L2-Wasserstein (W2) gradient flow coupled to non-conservative active pushing fields representing ATP-driven 3:2 Na+/K+ exchangers. We establish global weak existence of solutions to the continuous Poisson--Nernst--Planck--IB system via a split-step Jordan--Kinderlehrer--Otto (JKO) variational scheme, backed by Calderón--Zygmund bounds for non-local Poisson drift and H3(𝕊1) interface regularity under Stokes flow. We prove the continuous Rayleighian energy balance and establish exponential W2-stability of Non-Equilibrium Steady States (NESS) under a quantitative Bakry--Émery curvature threshold. Furthermore, we analyze the dual singular limits of kernel smoothing width w → 0 and dimensionless Debye length g = λD / L → 0. We establish the precise asymptotic regime connecting regularized and sharp models, prove the Γ-convergence of regularized energy functionals, derive distributional interface flux jump conditions, and resolve O(λD) Debye space-charge boundary layers to confirm bulk electroneutrality.
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BibTeX
@misc{lee2026immersed,
title = {{Immersed Boundary Electrodiffusion in the Wasserstein Gradient Flow Formalism: Global Well-Posedness, Non-Equilibrium Thermodynamics, and Singular Asymptotics}},
author = {Pilhwa Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40942},
eprint = {2610.40942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40942},
}
RIS
TY - UNPB
TI - Immersed Boundary Electrodiffusion in the Wasserstein Gradient Flow Formalism: Global Well-Posedness, Non-Equilibrium Thermodynamics, and Singular Asymptotics
AU - Pilhwa Lee
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40942
UR - https://arxiv.org/abs/2610.40942
AB - We introduce a unified optimal transport framework for the Immersed Boundary (IB) electrodiffusion active pump model. By representing membrane chemical potential barriers and active pump potentials via smoothed Eulerian kernels (Ψ_w, ℋ_w), we formulate the passive dynamics as an L^2-Wasserstein (W_2) gradient flow coupled to non-conservative active pushing fields representing ATP-driven 3:2 Na^+/K^+ exchangers. We establish global weak existence of solutions to the continuous Poisson--Nernst--Planck--IB system via a split-step Jordan--Kinderlehrer--Otto (JKO) variational scheme, backed by Calderón--Zygmund bounds for non-local Poisson drift and H^3(𝕊^1) interface regularity under Stokes flow. We prove the continuous Rayleighian energy balance and establish exponential W_2-stability of Non-Equilibrium Steady States (NESS) under a quantitative Bakry--Émery curvature threshold. Furthermore, we analyze the dual singular limits of kernel smoothing width w → 0 and dimensionless Debye length g = λ_D / L → 0. We establish the precise asymptotic regime connecting regularized and sharp models, prove the Γ-convergence of regularized energy functionals, derive distributional interface flux jump conditions, and resolve O(λ_D) Debye space-charge boundary layers to confirm bulk electroneutrality.
ER -