WIAS Preprint No. 2807, (2021)

Global existence analysis of energy-reaction-diffusion systems


  • Fischer, Julian
    ORCID: 0000-0002-0479-558X
  • Hopf, Katharina
    ORCID: 0000-0002-6527-2256
  • Kniely, Michael
    ORCID: 0000-0001-5645-4333
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

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.

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