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

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 and global existence of strong solutions in the class $\mathrm{W}^{(1,2)}_p(J \times \Omega; \mathbb{R}^N)$, and of classical H\"older solutions in the class $\mathrm{C}^{(1+\alpha, 2 + 2\alpha)}(\overline J \times \overline{\Omega})$. Exploiting that the model is based on thermodynamic principles, we further show a priori bounds related to mass conservation and the entropy principle.


Introduction
In chemical engineering catalytic processes often play an important, if not predominant role: Certain chemical reactions taking place within a chemical reactor are supposed to be accelerated, whereas other, usually undesirable, side reactions should be suppressed. This aim can be accomplished by adding substances which catalytically act in the fluid mixture (homogeneous catalysis), or e.g. by using suitable structures for the reactor surface (active surface) which may then act as a catalytic surface to accelerate the desirable reactions on the surface. In many cases such heterogeneous catalysis mechanism are actually more efficient by several orders of magnitude than homogeneous catalysts, and, moreover, one may often avoid the need for filtration technology to separate the desired product from the catalyst. Heterogeneous catalysis mechanisms and sorption processes may be modelled starting from a continuum thermodynamic viewpoint by reaction-diffusion systems in the chemical reactor and on the active surface which are coupled via sorption processes, i.e. the exchange of mass between the boundary layer of the bulk phase and the active surface, cf. [21]. In accord with their purpose, catalytic accelerated chemical reactions on the surface are very fast, i.e. both the surface chemistry (at least for the desired reactions) as well as sorption processes take place on very small time scales. Hence, it is natural to consider limit models, for which the surface chemistry and sorption are taken to be infinitely fast, i.e. surface chemistry and sorption processes are modelled as if they would attain an equilibrium configuration instantaneously. Using a dimensionless formulation of such coupled reaction-diffusion-sorption bulk-surface systems, several of such limit models have been proposed in [4], including a general formulation of such a fast sorption and fast surface chemistry model. The mathematical analysis of such systems there has been accomplished for the case of a three-component system with chemical reactions of type A Σ 1 + A Σ 2 ⇋ A Σ 3 on the surface, neglecting any bulk chemistry (the latter being no strong obstacle, and for the construction of (uniquely determined) strong solutions not a highly relevant assumption). In the present manuscript, the mathematical analysis is continued for limit systems of the same structure, but for general bulk and surface chemistry. In particular, the results on local-in-time existence of strong solutions, positivity, first blow-up criteria as well as a-priori estimates for the solutions will be extended to the generic case. The paper is organised as follows: In Section 2 some basic notation is introduced and some preliminary results are recalled. Thereafter, in Section 3, the class of heterogeneous catalysis models considered in this manuscript is introduced and the underlying modelling assumptions recalled from the article [4]. Section 4 constitutes the core of this article, and is splitted into subsections on L p -maximal regularity of a linearised version of the fastsorption-fast-surface-chemistry model, on local-in-time existence of strong W (1,2) p -solutions, and on an abstract blow-up criterion as well as a-priori bounds, e.g. entropy estimates, on the strong solution. The appendix then serves to sketch how to generalise the main results of [8] and [9] to the slightly more general setup considered here, where the system splits into parts on which Dirichlet type conditions, i.e. boundary conditions prescribed by differential operators of order zero, are imposed, and parts on which a no-flux boundary condition, i.e. given via a differential operator of order one, is used. There is a vast amount of literature on reaction-diffusionsystems or general parabolic systems in the bulk phase with homogeneous or inhomogeneous, linear or nonlinear boundary conditions, e.g. [16], [19] for a start, and quite recently thermodynamic principles have become a resourceful driving force for entropy methods, e.g. [12], [13], [14]. Astonishingly, however, up to now (at least to our knowledge) combined type boundary conditions, i.e. systems where at a fixed boundary point z ∈ ∂Ω Dirichlet type boundary conditions are imposed on some of the variables (or, a linear combination thereof), whereas the remaining variables satisfy Neumann type boundary conditions, have not been considered in the literature yet. Instead, other types of generalisations have been considered so far: Several authors, e.g. [1], [18], [22], did indeed consider parabolic systems with nonlinear boundary conditions, but these are always assumed to be of a common order, cf., e.g., the non-tangentiality condition in [18]. In, e.g., [11], [6] and [20] general parabolic systems or reaction-diffusion-systems with dynamic boundary conditions have been considered, i.e. typically two parabolic systems in the bulk phase and on the surface are coupled. In [10], on the other hand, the authors consider more general structures leading to parabolic systems based on the notion of a Newton polygon.
Remark 2.1 (Sobolev-Slobodetskii spaces and Besov spaces). Recall that for sufficiently regular domains Ω ⊆ R n , one has B s pp (Ω) = W s p (Ω) for s ∈ R + \ N 0 , but B k p (Ω) = W k p (Ω) for k ∈ N. For the definitions, basic properties and more information on these spaces, the interested reader is referred to the vast literature, e.g. [17], [2] and [3]. Most importantly for our purposes, the following embedding results hold true.

The model
In this paper, the following, rather general fast sorption, fast surface chemistry limit model will be considered: where the appearing variables and coefficients have the following physical interpretation and relations with each other. Thermodynamic and geometric variables and vectors: • c = (c 1 , . . . , c N ) T : R ×Ω → R N denotes the vector field of molar concentrations, i.e. c i (t, z) ∈ R is the molar concentration of the chemical substance A i at time t ∈ R in position z ∈ Ω, for i = 1, . . . , N ; • J = [j 1 · · · j N ] : R ×Ω → R n×N for j i : R ×Ω → R n the vector field of individual mass fluxes of species A i , i = 1, . . . , N ; • n : ∂Ω → R n , the outer normal vector field to Ω on ∂Ω; • r(c) = m a=1 R a (c)ν a , the vector field of total molar reaction rates in the bulk phase, modelling chemical reactions given by the formal (reversible) chemical reaction equations where α a = (α a 1 , . . . , α a N ) T , β a = (β a 1 , . . . , β a N ) T ∈ N N 0 and the stoichiometric vector of the a-th reaction is given by ν a = β a − α a ∈ Z N . Moreover, R a (c) denotes the effective reaction rate for the a-th chemical reaction in the bulk phase; • ν Σ,a = β Σ,a −α Σ,a ∈ Z N , a = 1, . . . , m Σ , are the stoichiometric vectors of the surface chemical reactions where α Σ,a = (α Σ,a 1 , . . . , α Σ,a N ) T , β Σ,a = (β Σ,a 1 , . . . , β Σ,a N ) T ∈ N N 0 for the adsorbed versions A Σ i of species A i .

Modelling assumptions:
• The concentrations are assumed to be very small (cp. to a reference concentration c ref of some solute which is not included in the model, 0 ≤ ci(t,z) /c ref ≪ 1, dilute mixture), and the fluid in the bulk is at rest, so that Fickian diffusion, is a reasonable (though, not thermodynamically consistent) model for the diffusive fluxes; • the chemical potentials µ i in the bulk phase are modelled as an ideal mixture with corresponds to some temperature-dependent equilibrium configuration and x i = ci c the scalar field of molar fractions, where c = N +1 i=1 c i is the total concentration density in the bulk phase, including the concentration of some solvent A N +1 . Instead of including the solvent A N +1 in the model, we replace c by some constant approximation c ref to the actual total concentration c, so that we may consider the vector x = (x 1 , . . . , x N ) T = c /c ref and its dynamics instead of c. Formally assuming c ref = 1, we then have µ i (c, ϑ) = µ 0 i (ϑ) + ln c i . Additionally, an isothermal system is assumed, hence µ 0 i (ϑ) = µ 0 i ∈ R is simply a constant; • the reaction velocity R a (c) of the a-th reaction is modelled (consistent with the entropy law) as • Throughout, we assume that all equilibria of the surface chemistry are detailed-balanced, i.e. ν Σ,1 , . . . , ν Σ,m Σ are linearly independent.
Then e k ∈ R N , k = 1, . . . , n Σ := N − m Σ , denotes a set of linearly independent conserved quantities under the surface chemistry, spanning the orthogonal complement of the surface chemistry stoichiometric vectors {ν Σ,a : a = 1, . . . , m Σ }. Under these assumptions, and the additional assumption that the sorption processes and surface chemistry take place very fast, i.e. on much smaller time scales than the bulk diffusion and the bulk chemistry, it has been demonstrated in [4] that (GFLM) is a reasonable limit model for the limiting case of infinitely fast surface chemistry and sorption processes (actually, independent of whether one of these two fast thermodynamic mechanism is even ultra-fast), and the condensed form of the limit model (including initial data) reads where Σ = ∂Ω denotes boundary of Ω, acting as an active surface, D = diag(d 1 , . . . , d N ) is the diagonal matrix of (Fickian) diffusion coefficients, and some initial data c 0 : Ω → R N are given.

Local-in time well-posedness for arbitrary bulk and surface chemistry
This section is devoted to the local-in time well-posedness analysis for generic fast-sorption-fast-surfacechemistry limit models of the form (1)- (4).
Since, by the detailed-balance assumption on the surface chemistry, the stoichiometric vectors ν Σ 1 , . . . , ν Σ m Σ are linearly independent, this system is equivalent to the system Multiplying the latter equation by exp(−α Σ,a · µ| Σ ) gives the formulation Here, we insert the special choice µ i = µ 0 i + ln c i for the bulk chemical potentials to obtain the system for the equilibrium constant κ a = exp(−ν Σ,a · µ 0 ). A possible linearised (around some c * : Ω → (0, ∞)) version of this nonlinear system is obtained by taking the partial derivatives ∂ or, for short Remark 4.1. Since only concentrations c i , c Σ i ≥ 0 have physical significance, only linearisations around states for which all components are (uniformly) strictly positive, i.e. only uniformly strict positive initial data, will be feasible by this approach towards linearisation. Regularisation effects for reaction-diffusion systems (cf. the strict parabolic maximum principle), however, suggest that this is no severe restriction as (under slight structural assumptions on the reaction network) for any initial data c 0 i ≥ 0, but not identically zero, the solution immediately becomes strictly positive, cf. the strict maximum principle for reaction-diffusion equations.
The program of the remainder of this section is the following: • Show L p -maximal regularity of the linearised problem, provided sufficient regularity of the reference function c * . This can be done based on abstract theory in a slightly extended version of the results in [8] and [9], so that mainly the validity of the Lopatinskii-Shapiro condition and regularity properties have to be checked. • Use L p -maximal regularity of the linearised problem and the contraction mapping principle to establish local-in-time existence for the fast-sorption-fast-surface-chemistry limit, provided the initial data are regular enough, have uniformly strictly positive entries and satisfy suitable compatibility conditions. • Moreover, this procedure will give a "natural" blow-up criterion for global-in time existence, where by "natural" it is meant that this norm appears in the contraction mapping argument for the local-in time existence.
4.1. L p -maximal regularity for the linearised problem. To show that the linearised problem (LP) possesses the property of L p -maximal regularity, let us first consider the case of constant coefficients (i.e. a constant reference function c * ∈ (0, ∞) N ) and a flat boundary (i.e. consider the special case of a boundary Σ = R n−1 ×{0} for the half-space domain Ω = R n−1 ×(0, ∞)). The corresponding linear initial-boundary value problem to be investigated, on the half-space then takes the general form of a parabolic reaction-diffusion system with boundary conditions of inhomogeneous type. For technical reasons (unboundedness of the domain Ω) shifted by a damping constant µ ≥ 0, hence, it reads as where one writes z = (z ′ , z n ) ∈ R n−1 × R + for the spatial variables, and the right hand sides -as the analysis of the linearised problem will reveal -have to satisfy the following regularity conditions: The corresponding maximal regularity result on the half-space reads as follows: Proposition 4.2 (L p -maximal regularity on the half-space for constant coefficients). Assume that c * ∈ (0, ∞) N is a constant and let p ∈ (1, ∞). Then there is µ 0 ≥ 0 such that for all µ ≥ µ 0 , the half-space problem (6) admits a unique solution v ∈ W , and, moreover, the following compatibility conditions are satisfied, if the respective time traces exist: In this case, there is C = C(p, µ) > 0, independent of the boundary and initial data, such that Proof. Starting with the system of PDEs (6), taking the partial Laplace-Fourier transform F for (λ, ξ ′ ) ∈ C + 0 × R n−1 , and setting y = z n , leads (formally) to the following parameter-dependent initial value problems As we look for a solution in the class v i = F −1 (v i ) ∈ L p ((0, T ); L p (R n−1 ×(0, ∞)), one necessarily needs to havê v i,+ (λ, ξ ′ ) = 0 for a.e. (λ, ξ ′ ) ∈ C + 0 × R n−1 , and in that case To match the solution with the boundary conditions at Σ = R n−1 ×{0}, we thus need to solve the linear systems Since all matrices D, R and C a are diagonal and the matrices ..,N up to a non-zero factor (c * | Σ ) ν Σ,a this matrix is invertible if and only if  ii ∈ C + 0 on the diagonal. Due to Lemma 4.3 below, this is the case. Using standard properties of the partial Fourier-Laplace transform and suitable multiplier theorems, it now follows that there is µ 0 ≥ 0 such that for all µ ≥ µ 0 , the system (6) has a unique solution in the class v ∈ W Let δ j ∈ C, j = 1, . . . , N , be such that 0 ∈ conv{δ j : j = 1, . . . , N } and the matrix M ∈ C N ×N be defined as Then M is invertible, i.e. Dv 1 , . . . , Dv m , w 1 , . . . , w s form a basis of C N .
Proof. As M ∈ C n×n is a square matrix, it suffices to demonstrate injectivity of M . Let u ∈ N(M ). Then, in particular, In particular, since V T = V * (as V has real entries), for the inner product on C m one finds As 0 ∈ conv{δ i : i = 1, . . . , m}, and |(V γ) i | 2 ≥ 0, this can only hold true if u = V γ = 0, and M must be injective.
The general L p -maximal regularity theorem (for bounded C 2 -domains Ω) can then be derived via the standard technique, i.e. first a generalisation to the bend space problem and, thereafter, a localisation procedure. For these techniques to work properly, one needs additional conditions on the (then non-constant) reference function c * : Ω → R N . In this particular case, the bulk diffusion operator −D∆ does not depend on the spatial position z ∈ Ω. Therefore, there is no need to consider perturbations of it, i.e. A sm = 0 in the language of [8], [9]. Neither do the conserved quantities e k , k = 1, . . . , n Σ , but only the matrix C(z) = diag(c * | Σ ) −1 (z) (which is C a up to a a-dependent factor (c * | Σ ) ν Σ,a ) depend on the spatial position z ∈ Ω. Using the same strategy as in [8], one may write where C sm (z) corresponds to a small perturbation of C(z 0 ). As in [8], one then considers the problem one then needs to derive a fixed point equation of the form The regularity assumptions in [9] now suggest that one should demand the following regularity of C, hence of c * : There are s, r ≥ p with 1 s + n−1 hence the reference function c * should be uniformly positive and Remark 4.4. Note that for regularity of C, one has to consider the regularity of the functions (c * k ) −1 , hence of ∂ ∂t Since the reference function should generally lie in the function class c * ∈ W , one naturally should take s = r = p, and then the condition on the values of s and r reads n + 1 2p Note that in this case the embeddings W if and only if the data are subject to the following regularity and compatibility conditions: ( . In this case, the solution depends continuously on the data, i.e. for some C = C(n, Ω) > 0 independent of the data it holds that As for many semi-linear systems, L p -maximal regularity, and in this case L p -L q -estimates can be employed to find a (unique) strong solution of the quasi-linear problem. This will be the aim of the next subsection.

4.2.
Local-in time existence of strong solutions, blow-up criteria and a-priori bounds. Having maximal regularity for the linearised problem at hand, we may now come back to the non-linear problem. L p -maximal regularity will play the typically crucial role in the construction of strong solutions via the contraction mapping principle. Thereby, rather as a by-product, a condition for continuation of the solution to a maximal solution will be established, where in general the solution may be global in time (which might be expected for a large subclass of the systems considered here), or one may observe a blow-up in finite time. Whereas the boundedness of the B 2−2/p pp -norm cannot be guaranteed in general, or more precisely, it is unclear whether boundedness holds true without any restriction on the bulk and surface chemistry, for slightly weaker norms a-priori bounds are possible, indeed. The latter will be considered in the second part of this subsection.

Local-in time existence and maximal continuation of solutions.
Recall the form of the fast-sorption-fastsurface-chemistry limit (5), and additionally consider a given initial datum c 0 : Ω → R N which should be regular enough (in a sense to be made precise later on): Introducing v(t, z) := c(t, z) − c 0 (z) leads to the reaction-diffusion-sorption system for v as follows: and for τ > 0 consider the linear solution operator and w is the unique strong solution to the linear problem This problem can now be handled in the way typical for semi-linear parabolic problems, employing the maximal regularity of the linearised problem and the regularity of the nonlinear maps on the right hand side, which allows for a fixed point argument via the contraction mapping principle. To establish the regularity properties which are needed, one first needs the following auxiliary result on algebraic properties of W the embedding constant constants for the continuous embeddings can be chosen independently of T ∈ (0, T 0 ), e.g.
Proof. Since for u ∈ D 0 (T ) one has u(0) = 0, it follows that From here the assertion follows easily. 2 and assume that Ω ⊆ R n is a bounded domain of class ∂Ω ∈ C 2 . Then the fast-sorption-fast-surface-chemistry limit problem (5') admits a unique local-in-time strong solution, if More precisely, for every reference initial datum c * 0 ∈ I + p (Ω), there are T > 0, ε > 0 and C > 0 such that the following statements hold true: (2) For any two initial data c 0 , (3) Any strong solution c ∈ W (1,2) p (J × Ω) can be extended in a unique way to a maximal strong solution Proof. Let ε > 0 and initial data Then, v should be solution to the following initial-boundary value-problem: in Ω.
The nonlinear boundary conditions can be expressed (using that (c 0 ) ν Σ,a = κ a ) as in Ω.
Therefore, v ∈ D ρ,T is a solution to ( * ), if and only if v ∈ D ρ,T is a fix point of the map Φ : D ρ,T → W (1,2) p ((0, T )× Ω; R N ) defined as follows: For v ∈ D ρ,T let Φ(v) := w be the unique solution to the inhomogeneous initialboundary value-problem By L p -maximal regularity of the linearised problem, the solution w ∈ W (1,2) p ((0, T ) × Ω; R N ) exists and is uniquely determined, for every v ∈ D ρ,T . I.e. Φ is well-defined. Since the initial datum w(0, ·) = 0 is zero for all the functions constructed in this way, the constant C T = C T0 > 0 in the maximal regularity estimate for v ∈ D ρ,T can be chosen independently of T ∈ (0, T 0 ] (using, e.g. a mirroring argument). We will demonstrate that ρ ∈ (0, ρ 0 ] and T ∈ (0, T 0 ] can be chosen such that Φ is a contractive self-mapping on D ρ,T , and hence attains a unique fixed point by the contraction mapping principle. To this end, we first show that Φ is for ρ ∈ (0, ρ 0 ] and suitable T ∈ (0, T 1 ] a mapping from D ρ,T into D ρ,T . We will employ the notation The term r(c * + v) Lp can be handled analogously to reaction-diffusion-systems with linear boundary conditions as where C(ρ, T ) → 0 as T → 0+ for ρ ∈ (0, ρ 0 ]. Next, which tends to zero as T → 0+. Moreover, -the former factor can be estimated as which can be estimated as can be handled using the estimate also with a constant C ρ,T tending to zero as T → 0+ and ρ ∈ (0, ρ 0 ]. These considerations give that T 1 ∈ (0, T 0 ] may be chosen such that for all ρ ∈ (0, ρ 0 ] and T ∈ (0, T 1 ], the map Φ maps D ρ,T into itself. Thus, it remains to show that Φ is contractive for suitable choice of ρ, T from this range. Let, therefore, ρ ∈ (0, ρ 0 ] and T ∈ (0, T 1 ] as well as u, v ∈ D ρ,T be given. With L p -maximal regularity and w(0, ·) = 0 it holds again Similar to the self-mapping property, the bulk chemistry term does not pose much problems and can be handled in an analogous way as above. For the termĥ(c * , v, u), we derive the following estimates: It holds where the latter term is and may be estimated as where C ρ,T → 0 as T → 0+, ρ ∈ (0, ρ 0 ]. The first term may be estimated writing can be estimated as can be handled analogously: Each of the summands can in W p for some C ρ,T tending to zero as T → 0+. As a result, choosing ρ ∈ (0, ρ 0 ] arbitrary and T ∈ (0, T 1 ] sufficiently small, say T ∈ (0, T 2 ] for some T 2 ∈ (0, T 1 ], Φ : D ρ,T → D ρ,T is a strict contraction, and, therefore, by the contraction mapping principle has a unique fixed point v * = Φ(v * ) ∈ D ρ,T . Then, c := c * + v * is the unique solution of the fast sorption and fast surface chemistry limit reaction diffusion system. Moreover, for c * ,0 ∈ I ε p (Ω), η > 0 sufficiently small and initial data c 0 ∈ B η (c * ,0 ) ⊆ I ε p (Ω) close to c * ,0 , there is a common choice of parameters ρ 0 and T 2 to make the respective maps Φ = Φ c 0 strictly contractive self-mappings for any ρ ∈ (0, ρ 0 ] and T ∈ (0, T 2 ], so that for all these initial data the solution exists and is unique at least on the time interval (0, T 2 ). Also it can be seen that as the maps Φ c 0 continuously depend on the initial datum, so do the fixed points, hence the solutions to the fast sorption and fast surface chemistry reaction-diffusion-limit system.
From the proof one can extract blow-up criteria for solutions which are not global in time.
Remark 4.9. The inclusion of the case min c i (t, z) → 0 for some i seems to be a bit out of place here, but is due to the chosen linearisation around the reference function. To have enough regularity for C = diag(c| Σ ) −1 one needs uniform positivity of the solution candidate c, thus on the initial datum c 0 . Therefore, this approach breaks down as min c i (t, ·) → 0.

4.2.2.
A-priori bounds on the strong solution of the fast sorption-surface-chemistry-transmission model. In the previous subsection, it has been noticed that a bound on the phase space norm · B 2−2/p pp is enough for establishing global existence of a strong solution. To derive such a bound, however, is a delicate matter, and it is not clear whether global existence holds true in all cases. On the other hand, for some weaker norms at least a-priori bounds can be established for free. The derivation of these a-priori bounds is based on the parabolic maximum principle and entropy considerations, highlighting the fruitful interplay between mathematics and physics, and will be presented in this subsection.
Then, for every T 0 ∈ (0, T max ] ∩ R there is C = C(T 0 ) > 0, also depending on the initial data c 0 , such that the following a-priori bounds hold true: (1) L t ∞ L z 1 -a-priori estimate: sup where the constant can actually be chosen independent of c 0 and T 0 , but only depends on the ratio between the smallest and largest entry of e ∈ (0, ∞) N ; (2) L t 1 L z ∞ -a-priori estimate: sup z∈Ω c(·, z) L1([0,T0);R N ) ≤ C; (3) L t 2 L z 2 -a-priori estimate: c L2([0,T0)×Ω;R N ) ≤ C; (4) Moreover, the following entropy identity holds true: Proof. L t ∞ L z 1 -a-priori estimate: Since e ∈ (0, ∞) N is a conserved quantity for both the bulk and surface chemistry, r(c) · e = r Σ (c) · e = 0 for all values of c. Thus, using regularity properties of parameter-dependent integrals and the divergence theorem, for every t ∈ [0, T 0 ) it holds that and, since e ∈ (0, ∞) N , the map c → Ω |c · e| dz defines a norm which is equivalent to the standard L 1 -norm on the Lebesgue space L 1 (Ω; R N ). More precisely, establishing the first a-priori estimate for C = max i e i min i e i independent of T 0 > 0 and the initial datum c 0 . L t 1 L z ∞ -a-priori estimate: To derive the L 1 L ∞ -a-priori bound, let us consider the function w : As a parameter integral of a C (1,2) -function, w has the regularity w ∈ C 2 ([0, T 0 ) × Ω) and using elementary results on parameter-dependent integrals, the evolution equation (5) and the assumption that e is a conserved quantity, we establish the estimate Therefore, w ≥ 0 satisfies the system of differential inequalities From the parabolic maximum principle for differential inequalities, it then follows that there is C > 0 (depending on T 0 and c 0 ) such that 0 ≤ w(t, z) ≤ C, t ∈ [0, T 0 ), z ∈ Ω and, consequently, one finds that L t 2 L z 2 -a-priori estimate: For the L 2 -estimate, let us fix T ∈ (0, T 0 ). Employing integration by parts, Fubini's theorem, the no-flux boundary conditions on the conserved part and the fundamental theorem of calculus, we find for the integral Dc(s, z) · e ds dz dt Dc(s, z) · e ds dt dz D∂ n c(s, z) · e ds dσ(z) dt where in the last step it has been used that c ≥ 0 and, therefore, One may thus take Entropy identity: By the theorem on derivatives of parameter-integrals, and as the derivative of the function (0, ∞) ∈ x → x(ln x − 1) is ln x for all x ∈ (0, ∞), one finds that The assertion will be established if i (µ 0 i +ln(c i ))d i ∂ n c i dσ(z) = 0 can be proved. From the boundary conditions e k · ∂ n (Dc) = 0, there are scalar functions η a : [0, since µ i | Σ = µ Σ at all times t ≥ 0 and positions z ∈ Σ (sorption processes in equilibrium for the fast-sorptionfast-surface-chemistry limit model) and A Σ a = 0 at all times t ≥ 0 and all positions z ∈ Σ (chemical reactions on the surface in equilibrium). Therefore, this contribution to the sum vanishes, and the entropy identity follows by the fundamental theorem of calculus.

Appendix: L p -maximal regularity for parabolic and elliptic boundary value problems with boundary conditions of varying differentiability orders
General L p -maximal regularity and L p -L q -optimality results as in [8] and [9] for linear parabolic systems on bounded domains have been typically formulated for boundary conditions of the same type, so either on the Dirichlet data of the functions or on its normal flux through the boundary. These results are heavily based on harmonic analysis and multiplier theory on Banach spaces of class HT (UMD-spaces), see, e.g., [5], [7], [15]. In this manuscript, however, we encounter the situation where the boundary conditions have a combined type, i.e. on some parts of the vector of concentrations c, or, more precisely, some linear combinations of them corresponding to the stoichiometric vectors (nonlinear, or, in the linearised version, linear and inhomogeneous) Dirichlet data are prescribed, whereas on other parts, more precisely, on linear combinations corresponding to conserved quantities, no-flux boundary conditions have been imposed. For example, the L p -maximal regularity results in [8], Theorem 7.11 (half-space problem) and Theorem 8.2 (domains with compact boundary), rely on a version of the Lopatinskii-Shapiro condition in which all the boundary operators B j , j = 1, . . . , N , are homogeneous of degree m j ∈ N 0 and some perturbation argument. In the definition of a principle symbol as used there, e.g. in [8,Section 8.1], the principle part of the boundary operators considered here would not 'see' the Dirichlet type contribution, as their order of differentiation is 0, so strictly smaller than the order 1 for the Neumann type parts. In this appendix, therefore, some comments will be made on a possible extension of the results in [8] and [9] to boundary conditions of combined type. For this purpose, one has to closely follow the lines of [8, and [9] to observe which adjustments are needed to transfer the results there to this slightly more general setup. 5.1. Partial Fourier transforms. In [8, Subsection 6.1], the authors start from the elliptic boundary value problem on the half-space R n + := R n−1 ×(0, ∞) given as where A(D) and B j (D) only consist of principal parts for m ∈ N and m j ∈ {0, 1, . . . , 2m − 1} and with constant coefficients a α , b j,β ∈ B(E), and where E is some Banach space (which only later will be assumed to be of class HT , i.e. an UMD-space). Also, the standard . . . , α n ) T ∈ N n 0 is used. This makes the symbols homogeneous in ξ of order 2m for A and m j for B j , respectively. In the situation considered here, such a condition on the boundary operators is too restrictive. Therefore, we assume the following adjusted version concerning the operators B j instead: Assumption 5.1. The symbol A = |α|=2m a α ξ α is homogeneous of degree 2m, whereas for j = 1, . . . , m there are projections P j,k ∈ B(E), k = 0, 1, . . . , m j < 2m such that P j,k P j,k ′ = 0 for k = k ′ , and Moreover, we assume that the symbol A(D) is parameter-elliptic with angle of ellipticity φ A ∈ [0, π), i.e. (11) σ(A(ξ)) ⊆ Σ π−φ , ξ ∈ R n with |ξ| = 1 for some φ ∈ [0, π) and φ A ∈ [0, π) is the infimum of these φ. Here, one defines the sector Σ π−φ = {z ∈ C \{0} : |arg(z)| < π − φ} for φ ∈ [0, π). For p ∈ (1, ∞) and φ > φ A consider the boundary value problem (8)-(9) for given data λ ∈ Σ π−φ , f ∈ L p (R n+1 + ; E) and g k j ∈ W 2m−k p (R n+1 + ; R(P j,k )) for j = 1, . . . , m and k = 0, 1, . . . , m j < 2m. The function f can be extended trivially by zero to E 0 f ∈ L p (R n+1 ; E). Moreover, we set g j = mj k=0 g k j = mj k=0 P j,k g j (by invariance of R(P j,k ) under the projection P j,k )). A common strategy is the following: First, solve the inhomogeneous problem in the full space R n+1 , then consider the problem for w := u − P v which is homogeneous in R + n+1 , but inhomogeneous on the boundary ∂ R n+1 + . By this, the resulting function w : [8,Section 5]) and P is the restriction operator, restricting any function on R n+1 to a function on R n+1 + , has to solve the abstract boundary value problem . . , m, on ∂ R n+1 + , so that w.l.o.g. we may and will assume that f = 0 in (8)-(9) in the following. For functions u = u(z ′ , y) ∈ L 1 (R n+1 + ) ∩ L p (R n+1 + ) one may now introduce the partial Fourier transform in z ′ as This leads to the boundary value problem in the Laplace-Fourier space whereb jkl (ξ ′ ) = |γ|=l b j,(y,l) (ξ ′ )(ξ ′ ) γ . The ODE (12) may then be rewritten as a first order system by considering the E 2m -valued function for some ρ > 0. To exploit homogeneity of A(ξ), B j,k (ξ) later on, one further introduces the variables σ := λ ρ 2m , and b := ξ ′ ρ and defines a matrix-valued function A 0 (ξ ′ , λ) as . Note that the parameter-ellipticity of the symbol A(ξ) implies, in particular, the invertibility of a 0 (ξ ′ ) = a 0 . Just as in [8], the coefficients a j (ξ) are ξ ′ -homogeneous of degree j, and (12) for the special case f = 0 is equivalent to the first order in y system of ordinary differential equations (14) D y F v = ρA 0 (b, σ)F v, ξ ′ ∈ R n , y > 0.
In contrast to the situation in [8], however, the boundary symbols B j (ξ ′ ) = mj k=0 B j,k (ξ) are not homogeneous in ξ ′ of degree m j in general, but only the symbols B j,k (ξ) are homogeneous in ξ of order k, so that  0 (b, σ) is strictly separated by the real axis, i.e. σ (A 0 (b, σ))∩R = ∅ and iσ (A 0 (b, σ)) splits into two parts S ± (b, σ) which are strictly separated by the imaginary axis. More precisely, there are c 1 , c 2 > 0 such that Therefore, the spectral projections P ± (b, σ) onto S ± (b, σ) exist.

5.2.
The Lopatinskii-Shapiro condition on the half space. Similar to the situation in [8], the Lopatinskii-Shapiro condition for the half-space problem may be expressed as follows.
Assumption 5.2 (Lopatinskii-Shapiro condition). For each λ ∈ Σ π−φ and ξ ′ ∈ R n such that (λ, ξ ′ ) = (0, 0), the problem The proofs of [8, Proposition 6.2-6.4] then directly transfer to this slightly modified situation to give Proposition 5.3. Let A(D) be a parameter elliptic operator of order 2m and angle of ellipticity φ A ∈ [0, π). Let φ > φ A and λ ∈ Σ π−φ and assume that the Lopatinskii-Shapiro condition holds true. Then there exists a unique solution F u of the system (12)- (13), which is given by the sum where (·) 1 denotes the first component of an element in E 2m , and for some function M (b, σ) : E m → E 2m , which is jointly holomorphic in b and σ, and with where and Proof. The proposition is established in the following way: First, the linear problem is decomposed into three sub-problems, each of which is easier to handle than the original problem. The first of these is the problem (12)- (13) for the special case f = 0, i.e.
Secondly, one extends f by zero to E 0 f on the full space R n+1 and solves the elliptic problem on the full space which constitutes the solution F w 1 to the inhomogeneous problem . . , m. In the last step one returns to the first problem, but replaces g j by −B j (D)F w 1 , i.e. minus the boundary data for the solution to the second auxiliary problem on the full space: The only thing left to do there, is to derive a formula for the boundary values B j (D)w 1 on ∂ R n+1 + . As a first step, the case f = 0 may be handled analogously to [8,Proposition 5.3] which provides us with a unique solution (F v) 1 to the problem (8)-(9) for f = 0, and which has the form (18), where M (b, σ) is jointly holomorphic in the variables (b, σ). Since the projections P j,k project the problem to subspaces on which the corresponding system for the boundary operators is homogeneous in ξ ′ of degree k, uniqueness and existence of solutions, as well as the representation formula for the solution follow via the equivalent reformulation as a first order system. It, therefore, remains to prove holomorphy of the map M . This can be done almost literally as in the proof of [8, Proposition 6.2]: First, employing the closed graph theorem, M (z) is a linear and closed map E m → E 2m is uniformly bounded, first on the set D = B 1 (0) R n × Σ π−φ ⊆ R n × C, and then in a complex neighourhoodD of D, which is sufficiently close to D. Using the spectral projection P − and the Lopatinskii-Shapiro condition, one can then proceed by demonstrating continuity of v(z) = M (z)g, for any fixed g ∈ E m , on complex lines, also using that P − and the operators B 0 j,k are continuous. Thereof, complex differentiability follows rather easily. For the solution of the full space problem, one may employ the full space theory developed in [8,Section 5] Here, as before, we write E 0 for the trivial extension from R n+1 + to R n+1 , P for the restriction from R n+1 to R n+1 + , and A R n+1 for the L p (R n+1 )-realisation of the differential operator A(D). In the last of the three subproblems, one further employs a following representation of the boundary values F g B j (ξ ′ , 0) = (B j (D)F P (λ + Then, considerations as in the first step lead to the desired representation of (F w 2 ) 1 , and the proposition follows.
From this, we obtain the following kernel estimates.
Theorem 5.8. Let A(D) be a parameter elliptic operator of order 2m with angle of ellipticity φ A ∈ [0, π). Let φ > φ A . For j = 1, . . . , m let B j (D) be a boundary operator of order m j < 2m and of the form above. Assume that the Lopatinskii-Shapiro condition (LS) holds true. Let p ∈ (1, ∞), E be a Banach space, f ∈ L p (R n+1 + ; E) and g j ∈ mj k=0 P j,k W 2m−k p (R n+1 + ; E) for j = 1, . . . , m. Let λ ∈ Σ π−φ and let A be defined as above. Then there is a unique solution u ∈ W 2m−1 Moreover, u is given by and there is a constant C > 0 such that for 0 ≤ |α| ≤ 2m − 1 and λ ∈ Σ π−φ one has

H
B of the L p -realisation of the boundary value problem (34): Using the kernel estimates of Proposition 5.5, one may then prove that A B admits a bounded H ∞ -calculus, see the following analogue to [8,Theorem 7.4]: Corollary 5.11. Let E be a Banach space of class HT . Then there is C > 0 such that where R(M ) denotes the R-bound of a R-bounded set M .
5.6. R-bounds for solution operators. By Corollary 5.11, the solution map for the boundary value problem (34) with g = 0 admits R-bounds. Next, one has to consider the boundary value problem (34) for f = 0 and general g = 0. By Proposition 5.6, its solution may be expressed as v = m j=1 mj k=0 S j,k λ P j,k g j .
Lemma 5.13. There is λ 0 > 0 such that the set Proof. The proof is essentially the same as in [8,Section 7.3]. Instead of the operators B j (D), consider the operators B j,k (D) = B j (D)P j,k (k = 0, 1, . . . , m j ) and use that B j (D) = mj k=0 B j,k (D). 5.8. Variable coefficients in a half space. To generalise the results of the previous subsection to more general spatial variant coefficients, one may use the localisation procedure as in [8,Section 5.3]. In comparison with [8,Section 7.4] one has to adjust the notion of the principal part to a definition suitable for the combined type boundary conditions considered here. Throughout this subsection, assume that there are projections P j,k , j = 1, . . . , m, k = 0, 1, . . . , m j < 2m such that P j,k P j,k ′ = 0 for (j, k) = (j ′ , k ′ ) and assume that A and B j have the form A(z, D) = |α|≤2m a α (z)D α , B j (z, D) = mj k=0 |β|≤k b j,k,β (z)D β P j,k .
The procedure above then gives the following result.

5.9.
Localisation techniques for domains. In this subsection the statement of the maximal regularity theorem for domains will be formulated. Let E be a Banach space of class HT and m, n, m 1 , . . . , m m be natural numbers with m j < 2m. Moreover, let P j,k , j = 1, . . . , m, k = 0, . . . , m j , be projections such that P j,k P j,k ′ = 0 for (j, k) = (j ′ , k ′ ). Let Ω ⊆ R n+1 be a domain with C 2m -boundary, i.e. for each z 0 ∈ ∂Ω local coordinates exist which are obtained from the original one by rotating and shifting, and such that the positive x n+1 -axis corresponds to the direction of the outer normal vector n to Ω at z 0 . The local coordinates may be chosen such that they depend continuously on z 0 ∈ ∂Ω. Consider the differential operators (1) Smoothness conditions: (a) a α ∈ C l (Ω; B(E)) for each |α| = 2m; (b) a α ∈ [L ∞ + L r k ](Ω, B(E)) for each |α| = k < 2m with r k ≥ p and 2m − k > n r k ; (c) b j,β ∈ C 2m−k (∂Ω; B(E)) for each j = 1, . . . , m, 0 ≤ k ≤ m j and |β| = k.
Proof. The localisation procedure as presented in [8, Section 8.2] carries over almost literally; again one has to consider the operators B j (D)P j,k for each pair (j, k) separately, but the technique stays the same.