Global existence analysis of energy-reaction-diffusion systems

TL;DR

This paper proves the global existence of solutions for complex energy-reaction-diffusion systems that incorporate thermodynamic principles, cross-diffusion effects, and species-dependent diffusivities, using advanced mathematical techniques.

Contribution

It introduces a rigorous mathematical framework for analyzing thermodynamically consistent, non-isothermal reaction-diffusion systems with cross-diffusion effects and species-dependent diffusivities.

Findings

Established global-in-time existence of solutions.

Developed a priori estimates based on entropy methods.

Handled non-integrable fluxes with renormalised solutions.

Abstract

We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.