Fast reaction limits and convergence rate for nonlinear bulk-surface reaction-diffusion systems modeling reversible chemical reactions

TL;DR

This work addresses the fast-reaction limit for a nonlinear bulk-surface reaction-diffusion system modeling reversible chemical reactions with arbitrary $α,β≥1$. It develops a rigorous framework using a product-space Aubin–Lions argument and entropy methods to prove strong convergence to a heat equation in the bulk coupled to a nonlinear dynamical boundary condition on the surface, with the limit satisfying $u^α|_{Γ}=v^β$. For $α=β$, it further establishes a convergence rate of order $O(\, oot 2 ext of{ε} ext{)}$ under a compatibility condition on the initial data. Collectively, the results extend known linear bulk-surface fast-reaction limits to the nonlinear regime and provide a rigorous bridge to effective boundary-dynamics models relevant for coupled bulk-surface kinetics.

Abstract

The fast reaction limit for a nonlinear bulk-surface reaction-diffusion system is investigated. The system model describes a reversible reaction with arbitrary stoichiometric coefficients, where one chemical is present in a bounded vessel $Ω$ and the other chemical lies only on the boundary $\partialΩ$ where the reaction takes place. In the limit as the reaction rate constant tends to infinity, we prove that the solution converges strongly to that of a heat equation with nonlinear dynamical boundary condition. This is obtained by showing a-priori estimates of solutions which are uniform in the reaction rate constants. In order to overcome the difficulty caused by the bulk-surface coupling, we consider the limit in suitable product spaces where the Aubin-Lions lemma is applicable. Moreover, in the case of equal stoichiometric coefficients, we obtain the convergence rate of the fast reaction limit by exploiting suitable estimates of the limiting system.

Theorems & Definitions (26)

• Definition 2.1
• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof