Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness

TL;DR

This paper extends the concept of renormalised solutions to reaction-diffusion systems with nonlinear interface conditions, establishing global existence and stability results for entropy-dissipating models with complex interfacial transmission rates.

Contribution

It introduces a novel framework for renormalised solutions in coupled reaction-diffusion systems with nonlinear interfaces, enabling analysis without growth restrictions.

Findings

Proved global existence of solutions.

Established weak-strong stability estimates.

Handled power-law nonlinearities in interface conditions.

Abstract

We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy.