Analysis of Maxwell–Stefan systems for heat conducting fluid mixtures

Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

“…However, the question why, for certain situations, the Darcy law or other velocities can be substituted to momentum balance, seems not to have been yet completely answered for multicompenent fluids. Sometimes v = 0 has been postulated (See [DGM13], [Fuh15], [KZ16], [HJ21] or [Dru17a,Dru17b]) as the fastest way to come to reaction-diffusion systems, but often the only solution consistent with momentum balance is then the hydrostatic and chemical equilibrium. In non-equilibrium, the Navier-Stokes equations cannot be simultaneously fulfilled.…”

Nonisothermal Richards flow in porous media with cross diffusion

“…[10,24,25]), but the focus on how to model cross-diffusion effects (for instance in fluid mixtures) in a thermodynamically consistent way attracted the mathematical community's attention only recently, see e.g. [7,8,9,13,15,18,22,26]. For seminal works on the thermodynamic theory of fluid mixtures we want to mention e.g.…”

“…A recent breakthrough was made by Buliček, Jüngel, Pokorný and Zamponi in [8] by studying a steady compressible thermodynamically consistent Navier-Stokes-Fourier system for chemically reacting fluids with cross-diffusion effects. Very recent results were performed in [22], where the Maxwell-Stefan-Fourier equations in Fick-Onsager form were investigated and in [18], where a thermodynamically consistent reaction-cross-diffusion system coupled to an equation for the heat transfer was studied. The aim of this paper, namely to study in a thermodynamically consistent way the nonisothermal immiscible compressible two-phase unsaturated flow in porous media with cross diffusion, is completely new, to our best knowledge.…”

“…Here c w , c s > 0 are (scaled) heat capacities (of water and skeleton, respectively), p at > 0 is the atmospheric pressure, while (22) γ > 2.…”

Non-isothermal multicomponent flows with mass diffusion and heat conduction

“…Even though the Maxwell-Stefan system has been studied extensively in the isothermal case, the only known works in the non-isothermal case concerns the local-in-time existence and uniqueness of classical solutions in [11] and the global-in-time existence of weak solutions in [8]. The goal in [5] is to obtain global-in-time weak solutions for the above Maxwell-Stefan-Fourier system, that is compatible with thermodynamics and differs from the model presented in [8] in several points, as explained in [5, Section 1]. Theorem 3.…”