---
_id: '10547'
abstract:
- lang: eng
  text: "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,\r\nwhile 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."
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).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
- first_name: Katharina
  full_name: Hopf, Katharina
  last_name: Hopf
- first_name: Michael
  full_name: Kniely, Michael
  id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
  last_name: Kniely
  orcid: 0000-0001-5645-4333
- first_name: Alexander
  full_name: Mielke, Alexander
  last_name: Mielke
citation:
  ama: Fischer JL, Hopf K, Kniely M, Mielke A. Global existence analysis of energy-reaction-diffusion
    systems. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(1):220-267. doi:<a
    href="https://doi.org/10.1137/20M1387237">10.1137/20M1387237</a>
  apa: Fischer, J. L., Hopf, K., Kniely, M., &#38; Mielke, A. (2022). Global existence
    analysis of energy-reaction-diffusion systems. <i>SIAM Journal on Mathematical
    Analysis</i>. Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/20M1387237">https://doi.org/10.1137/20M1387237</a>
  chicago: Fischer, Julian L, Katharina Hopf, Michael Kniely, and Alexander Mielke.
    “Global Existence Analysis of Energy-Reaction-Diffusion Systems.” <i>SIAM Journal
    on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics,
    2022. <a href="https://doi.org/10.1137/20M1387237">https://doi.org/10.1137/20M1387237</a>.
  ieee: J. L. Fischer, K. Hopf, M. Kniely, and A. Mielke, “Global existence analysis
    of energy-reaction-diffusion systems,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 220–267, 2022.
  ista: 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.
  mla: Fischer, Julian L., et al. “Global Existence Analysis of Energy-Reaction-Diffusion
    Systems.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1, Society
    for Industrial and Applied Mathematics, 2022, pp. 220–67, doi:<a href="https://doi.org/10.1137/20M1387237">10.1137/20M1387237</a>.
  short: J.L. Fischer, K. Hopf, M. Kniely, A. Mielke, SIAM Journal on Mathematical
    Analysis 54 (2022) 220–267.
date_created: 2021-12-16T12:08:56Z
date_published: 2022-01-04T00:00:00Z
date_updated: 2023-08-02T13:37:03Z
day: '04'
department:
- _id: JuFi
doi: 10.1137/20M1387237
external_id:
  arxiv:
  - '2012.03792 '
  isi:
  - '000762768000006'
intvolume: '        54'
isi: 1
issue: '1'
keyword:
- Energy-Reaction-Diffusion Systems
- Cross Diffusion
- Global-In-Time Existence of Weak/Renormalised Solutions
- Entropy Method
- Onsager System
- Soret/Dufour Effect
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2012.03792
month: '01'
oa: 1
oa_version: Preprint
page: 220-267
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  issn:
  - 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global existence analysis of energy-reaction-diffusion systems
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...