Analysis of some heterogeneous catalysis models with fast sorption and fast surface chemistry

Abstract

We investigate limit models resulting from a dimensional analysis of quite general heterogeneous catalysis models with fast sorption (i.e. exchange of mass between the bulk phase and the catalytic surface of a reactor) and fast surface chemistry for a prototypical chemical reactor. For the resulting reaction–diffusion systems with linear boundary conditions on the normal mass fluxes, but at the same time nonlinear boundary conditions on the concentrations itself, we provide analytic properties such as local-in-time well-posedness, positivity, a priori bounds and comment on steps towards global existence of strong solutions in the class $$\mathrm {W}^{(1,2)}_p(J \times \Omega ; {{\,\mathrm{{\mathbb {R}}}\,}}^N)$$ W p ( 1 , 2 ) ( J × Ω ; R N ) , and of classical solutions in the Hölder class $$\mathrm {C}^{(1+\alpha , 2 + 2\alpha )}({\overline{J}} \times {\overline{\Omega }}; {{\,\mathrm{{\mathbb {R}}}\,}}^N)$$ C ( 1 + α , 2 + 2 α ) ( J ¯ × Ω ¯ ; R N ) . Exploiting that the model is based on thermodynamic principles, we further show a priori bounds related to mass conservation and the entropy principle.

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

“…Examples and some discussions are to be found, a.o. in [AB21a,HMPW17,DDGG20], and of course in many other references.…”

“…We also refer to [AB21a,AB21b] for interesting discussions in the context of catalysis, or to [BP17]. For the heat flux, typical are for instance cooling conditions like −r Γ h (x, t, T ) = α (T − T ext (x, t)), where T ext is the given external temeprature and α a positive coefficient.…”

$\mathrm{L}_p$-maximal regularity for parabolic and elliptic boundary value problems with boundary conditions of mixed differentiability orders

“…More general results are then derived using operator-theoretic results from modern harmonic analysis and employing perturbation methods. In this manuscript, however, we want to cover situations similar to those in [4], where demanding homogeneity of the (adjusted notion of the) principle part of the boundary symbols cannot be guaranteed. Therefore, in comparison with [12], we relax the conditions on the principle parts of the boundary operators in the following way: Assumption 4.1.…”

“…I would like to thank Dieter Bothe for bringing up the interesting fast surface chemistry and fast sorption model considered in [3] and [4] and for encouraging me to extend the appendix of a preprint version of the latter to this full manuscript. Also, I would like to thank Robert Denk for his kind advice.…”

“…The proof for these results heavily employ harmonic analysis and multiplier theory on Banach spaces of class HT , see, e.g., Berksen and Gillespie [5], Clément, de Pagter, Sukochev and Witvliet [10], Kalton and Weis [22]. On the other hand, in a recent manuscript [4], the analysis of fast-surface chemistry and fast-adsorption limits for heterogeneous catalysis systems, led to a situation where the boundary conditions have a combined type: Instead of a Dirichtlet or Neumann condition in all components equally, a no-flux boundary condition occurred at certain linear combinations of the individidual normal fluxes, i.e., the normal direction of the gradient of the individual components, together with nonlinear boundary conditions at the boundary trace of the individual concentrations due to equilibrium conditions in extremely fast surface chemical reactions. Though that problem can be solved by emplying the results and techniques of Ladyshenskaya, Solonnikov and Uralceva [25], which are valid for finite-component systems of parabolic type, this problem stimulated the search for more abstract and more general results on vector-valued parabolic and elliptic boundary value problems in the spirit of [12] and [13].…”