Global existence analysis of energy-reaction-diffusion systems
- Fischer, Julian
- Hopf, Katharina
- Kniely, Michael
- Mielke, Alexander
2020 Mathematics Subject Classification
- 35Q79 35K51 35K57 80A19
- Energy-reaction-diffusion systems, cross diffusion, global-in-time existence of weak/renormalised solutions, entropy method, Onsager system, Soret/Dufour effect
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.
- SIAM J. Math. Anal., 54 (2022), pp. 220--267, DOI 10.1137/20M1387237 .