---
_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'
...
---
_id: '10005'
abstract:
- lang: eng
  text: We study systems of nonlinear partial differential equations of parabolic
    type, in which the elliptic operator is replaced by the first-order divergence
    operator acting on a flux function, which is related to the spatial gradient of
    the unknown through an additional implicit equation. This setting, broad enough
    in terms of applications, significantly expands the paradigm of nonlinear parabolic
    problems. Formulating four conditions concerning the form of the implicit equation,
    we first show that these conditions describe a maximal monotone p-coercive graph.
    We then establish the global-in-time and large-data existence of a (weak) solution
    and its uniqueness. To this end, we adopt and significantly generalize Minty’s
    method of monotone mappings. A unified theory, containing several novel tools,
    is developed in a way to be tractable from the point of view of numerical approximations.
acknowledgement: "M. Bulíček and J. Málek acknowledge the support of the project No.
  18-12719S financed by the Czech\r\nScience foundation (GAČR). E. Maringová acknowledges
  support from Charles University Research program \r\nUNCE/SCI/023, the grant SVV-2020-260583
  by the Ministry of Education, Youth and Sports, Czech Republic\r\nand from the Austrian
  Science Fund (FWF), grants P30000, W1245, and F65. M. Bulíček and J. Málek are\r\nmembers
  of the Nečas Center for Mathematical Modelling.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Miroslav
  full_name: Bulíček, Miroslav
  last_name: Bulíček
- first_name: Erika
  full_name: Maringová, Erika
  id: dbabca31-66eb-11eb-963a-fb9c22c880b4
  last_name: Maringová
- first_name: Josef
  full_name: Málek, Josef
  last_name: Málek
citation:
  ama: Bulíček M, Maringová E, Málek J. On nonlinear problems of parabolic type with
    implicit constitutive equations involving flux. <i>Mathematical Models and Methods
    in Applied Sciences</i>. 2021;31(09). doi:<a href="https://doi.org/10.1142/S0218202521500457">10.1142/S0218202521500457</a>
  apa: Bulíček, M., Maringová, E., &#38; Málek, J. (2021). On nonlinear problems of
    parabolic type with implicit constitutive equations involving flux. <i>Mathematical
    Models and Methods in Applied Sciences</i>. World Scientific. <a href="https://doi.org/10.1142/S0218202521500457">https://doi.org/10.1142/S0218202521500457</a>
  chicago: Bulíček, Miroslav, Erika Maringová, and Josef Málek. “On Nonlinear Problems
    of Parabolic Type with Implicit Constitutive Equations Involving Flux.” <i>Mathematical
    Models and Methods in Applied Sciences</i>. World Scientific, 2021. <a href="https://doi.org/10.1142/S0218202521500457">https://doi.org/10.1142/S0218202521500457</a>.
  ieee: M. Bulíček, E. Maringová, and J. Málek, “On nonlinear problems of parabolic
    type with implicit constitutive equations involving flux,” <i>Mathematical Models
    and Methods in Applied Sciences</i>, vol. 31, no. 09. World Scientific, 2021.
  ista: Bulíček M, Maringová E, Málek J. 2021. On nonlinear problems of parabolic
    type with implicit constitutive equations involving flux. Mathematical Models
    and Methods in Applied Sciences. 31(09).
  mla: Bulíček, Miroslav, et al. “On Nonlinear Problems of Parabolic Type with Implicit
    Constitutive Equations Involving Flux.” <i>Mathematical Models and Methods in
    Applied Sciences</i>, vol. 31, no. 09, World Scientific, 2021, doi:<a href="https://doi.org/10.1142/S0218202521500457">10.1142/S0218202521500457</a>.
  short: M. Bulíček, E. Maringová, J. Málek, Mathematical Models and Methods in Applied
    Sciences 31 (2021).
date_created: 2021-09-12T22:01:25Z
date_published: 2021-08-25T00:00:00Z
date_updated: 2023-09-04T11:43:45Z
day: '25'
department:
- _id: JuFi
doi: 10.1142/S0218202521500457
external_id:
  arxiv:
  - '2009.06917'
  isi:
  - '000722222900004'
intvolume: '        31'
isi: 1
issue: '09'
keyword:
- Nonlinear parabolic systems
- implicit constitutive theory
- weak solutions
- existence
- uniqueness
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2009.06917
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  eissn:
  - 1793-6314
  issn:
  - 0218-2025
publication_status: published
publisher: World Scientific
quality_controlled: '1'
scopus_import: '1'
status: public
title: On nonlinear problems of parabolic type with implicit constitutive equations
  involving flux
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2021'
...