Reaction–diffusion systems of Maxwell–Stefan type with reversible mass-action kinetics

Abstract

The mass-based Maxwell-Stefan approach to one-phase multicomponent reactive mixtures is mathematically analyzed. It is shown that the resulting quasilinear, strongly coupled reaction-diffusion system is locally well-posed in an Lp-setting and generates a local semiflow on its natural state space. Solutions regularize instantly and become strictly positive if their initial components are all nonnegative and nontrivial. For a class of reversible mass-action kinetics, the positive equilibria are identified: these are precisely the constant chemical equilibria of the system, which may form a manifold. Here the total free energy of the system is employed which serves as a Lyapunov function for the system. By the generalized principle of linearized stability, positive equilibria are proved to be normally stable.where the diffusive fluxes J k are given byNote that, by definition, k y k = 1 and k J k = 0.2000 Mathematics Subject Classification. 35R35, Secondary: 35Q30, 76D45, 76T10.

On the isothermal compressible multi-component mixture flow: The local existence and maximalLp−Lqregularity of solutions

“…When the species equations are decoupled from the fluid equations, the resulting system of PDEs is related to the Stefan-Maxwell system analyzed for example in [6,20]. In both of these papers the isobaric isothermal systems are considered with the barycentric velocity being equal to 0.…”

Exponential Time Decay of Solutions to Reaction-Cross-Diffusion Systems of Maxwell–Stefan Type

“…A number of articles is concerned with the large-time asymptotics in Maxwell-Stefan systems. For global existence results on these systems, we refer to [26,33,34]. In [33], the exponential decay to the homogeneous state state is shown with vanishing reaction rates and same molar masses.…”

“…The result was generalized to different molar masses in [9], but still without reaction terms. The convergence to equilibrium was proved in [22,Theorem 9.7.4] and [26,Theorem 4.3] under the condition that the initial datum is close to the equilibrium state. The work [26] also addresses the exponential convergence to a homogeneous equilibrium assuming (i) global existence of strong solutions and (ii) uniformin-time strict positivity of the solutions (see Prop.…”

Convergence of an implicit Euler Galerkin scheme for Poisson–Maxwell–Stefan systems

“…The entropy structure of the Maxwell-Stefan system was revealed in [21], and a general global existence theorem could be shown. Further global existence results can be found in [18,24]. The Maxwell-Stefan system was coupled to the heat equation [20] and to the incompressible Navier-Stokes equations [6].…”

“…The Maxwell-Stefan system was coupled to the heat equation [20] and to the incompressible Navier-Stokes equations [6]. In [15,Theorem 9.7.4] and [18,Theorem 4.3], the large-time asymptotics for initial data close to equilibrium was analyzed. The convergence to equilibrium for any initial data was investigated in [6,21] without production terms and in [7] with production terms for reversible reactions.…”