@article{10547,
  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.},
  author       = {Fischer, Julian L and Hopf, Katharina and Kniely, Michael and Mielke, Alexander},
  issn         = {0036-1410},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Energy-Reaction-Diffusion Systems, Cross Diffusion, Global-In-Time Existence of Weak/Renormalised Solutions, Entropy Method, Onsager System, Soret/Dufour Effect},
  number       = {1},
  pages        = {220--267},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Global existence analysis of energy-reaction-diffusion systems}},
  doi          = {10.1137/20M1387237},
  volume       = {54},
  year         = {2022},
}

@article{10005,
  abstract     = {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.},
  author       = {Bulíček, Miroslav and Maringová, Erika and Málek, Josef},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  keywords     = {Nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness},
  number       = {09},
  publisher    = {World Scientific},
  title        = {{On nonlinear problems of parabolic type with implicit constitutive equations involving flux}},
  doi          = {10.1142/S0218202521500457},
  volume       = {31},
  year         = {2021},
}