Global existence analysis of energy-reaction-diffusion systems

Fischer JL, Hopf K, Kniely M, Mielke A. 2022. Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 54(1), 220–267.


Journal Article | Published | English

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Author
Fischer, Julian LISTA ; Hopf, Katharina; Kniely, MichaelISTA ; Mielke, Alexander
Department
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.
Publishing Year
Date Published
2022-01-04
Journal Title
SIAM Journal on Mathematical Analysis
Acknowledgement
M.K. gratefully acknowledges the hospitality of WIAS Berlin, where a major part of the project was carried out. The research stay of M.K. at WIAS Berlin was funded by the Austrian Federal Ministry of Education, Science and Research through a research fellowship for graduates of a promotio sub auspiciis. The research of A.M. has been partially supported by Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1114 “Scaling Cascades in Complex Systems” (Project no. 235221301), Subproject C05 “Effective models for materials and interfaces with multiple scales”. J.F. and A.M. are grateful for the hospitality of the Erwin Schrödinger Institute in Vienna, where some ideas for this work have been developed. The authors are grateful to two anonymous referees for several helpful comments, in particular for the short proof of estimate (2.7).
Volume
54
Issue
1
Page
220-267
ISSN
IST-REx-ID

Cite this

Fischer JL, Hopf K, Kniely M, Mielke A. Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 2022;54(1):220-267. doi:10.1137/20M1387237
Fischer, J. L., Hopf, K., Kniely, M., & Mielke, A. (2022). Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1387237
Fischer, Julian L, Katharina Hopf, Michael Kniely, and Alexander Mielke. “Global Existence Analysis of Energy-Reaction-Diffusion Systems.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/20M1387237.
J. L. Fischer, K. Hopf, M. Kniely, and A. Mielke, “Global existence analysis of energy-reaction-diffusion systems,” SIAM Journal on Mathematical Analysis, vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 220–267, 2022.
Fischer JL, Hopf K, Kniely M, Mielke A. 2022. Global existence analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 54(1), 220–267.
Fischer, Julian L., et al. “Global Existence Analysis of Energy-Reaction-Diffusion Systems.” SIAM Journal on Mathematical Analysis, vol. 54, no. 1, Society for Industrial and Applied Mathematics, 2022, pp. 220–67, doi:10.1137/20M1387237.
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