Uniform convergence to equilibrium for a family of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall model

In this paper, we establish convergence to equilibrium for a drift–diffusion–recombination system modelling the charge transport within certain semiconductor devices. More precisely, we consider a two-level system for electrons and holes which is augmented by an intermediate energy level for electrons in so-called trapped states. The recombination dynamics use the mass action principle by taking into account this additional trap level. The main part of the paper is concerned with the derivation of an entropy–entropy production inequality, which entails exponential convergence to the equilibrium via the so-called entropy method. The novelty of our approach lies in the fact that the entropy method is applied uniformly in a fast-reaction parameter which governs the lifetime of electrons on the trap level. Thus, the resulting decay estimate for the densities of electrons and holes extends to the corresponding quasi-steady-state approximation.


Introduction and main results
The formulation and mathematical analysis of drift-diffusion type semiconductor models reach back to the middle of the last century, see e.g. [18,20,24] and the references therein; yet these models still form a highly relevant workhorse in the simulation of semiconductor devices and batteries.
Physically, drift-diffusion models describe the transport of charge carriers via diffusion and convection governed by electric fields. In semiconductors, charge carriers are electrons and holes (positively charged quasi-particles, which represent the absence of an electron). Pairs of electrons and holes can be "generated" and "destroyed" by recombination processes. Generation of an electron-hole pair occurs when an electron is lifted from a low-energy valence band to a high-energy conduction band, where electrons are mobile-leaving behind an equally mobile hole in the valence band. A pivotal generation-recombination model was formulated by Shockley, Read and Hall [16,21]. Mathematically, Shockley-Read-Hall recombination introduces quadratic non-linear reaction terms into the drift-diffusion dynamics.
A derivation of the Shockley-Read-Hall model considers a generation-recombination process as sketched in Fig. 1. It assumes that appropriately distributed foreign atoms in the crystal lattice of the semiconductor material facilitate the generation of electron-hole pairs by providing in-between energy levels, requiring smaller amounts of energy for each step. Since electrons are immobile at these in-between energy levels, they are called trapped states. Also their maximal density is limited. The Shockley-Read-Hall model of electron-hole recombination is obtained as a quasi-steady-state approximation of the trapped-state dynamics as detailed in the following.
We denote the charge densities of electrons, holes and trapped states by n, p and n tr and consider the following PDE-ODE drift-diffusion-recombination system: with the drift-diffusion fluxes and reaction terms (1) ⎧ ⎪ ⎨ ⎪ ⎩ t n = ∇ ⋅ J n (n) + R n (n, n tr ), t p = ∇ ⋅ J p (p) + R p (p, n tr ), t n tr = R p (p, n tr ) − R n (n, n tr ), The constants n 0 , p 0 , n , p > 0 are positive recombination parameters and ∈ (0, 0 ] for arbitrary 0 > 0 is a positive relaxation parameter. V n and V p represent external time-independent potentials. The reaction term R n models transitions of electrons from the trap level to the conduction band (proportional to n tr ) and vice versa (proportional to −n(1 − n tr ) ), where the maximum capacity of the trap level is normalised to one. The analogue processes with respect to the valence band are described by R p . Note that the rate of hole generation is equivalent to the rate of electrons moving from the valence band to the trap level, which is proportional to ( 1 − n tr ). Similarly, the annihilation of a hole corresponds to an electron that jumps from the trap level to the valence band, which yields a reaction rate proportional to −pn tr . Moreover, n 0 , p 0 > 0 represent reference levels for the charge concentrations n and p, while n , p > 0 are inverse reaction parameter. Note that the concentration of trapped states satisfies n tr ∈ [0, 1] provided this holds true for their initial concentration (cf. Theorem 1.1).
The dynamical equation for n tr in (1) is an ODE in time and pointwise in space with a right hand side depending on n, p and n tr via R n and R p . We stress that there is no drift-diffusion term for n tr since trapped electrons are immobile. This is due to the correlation between foreign atoms and the corresponding trap levels which are locally bound near these crystal impurities.
The parameter > 0 models the lifetime of trapped states, where lifetime refers to the expected time until an electron in a trapped state moves either to the valence or the conduction band. The Shockley-Read-Hall recombination model is obtained in the (formal) limit → 0 , where the concentration of trapped states is determined from the algebraic relation 0 = R p (p, n tr ) − R n (n, n tr ): In this quasi-steady-state approximation, the density of trapped states n tr and its evolution are eliminated from system (1), while the evolutions of the charge carriers n and p are subject to the Shockley-Read-Hall recombination terms J n ∶= ∇n + n∇V n = n ∇ n n , n ∶= e −V n , A rigorous proof of this quasi-steady-state approximation has been performed in [15], even for more general models. See also [18] for semiconductor models with reaction terms of Shockley-Read-Hall-type.
We complete the mathematical description by considering system (1) on a bounded domain Ω ⊂ ℝ m , m ≥ 1 , with sufficiently smooth boundary Ω . Without loss of generality, we suppose that the volume of Ω is normalised, i.e. |Ω| = 1 , which can be achieved by an appropriate scaling of the spatial variables.
We impose no-flux boundary conditions for J n and J p , where n denotes the outer unit normal vector on Ω , and we prescribe non-negative and bounded initial data n I , p I , n tr,I ∈ L ∞ (Ω) together with ‖n tr,I ‖ L ∞ (Ω) ≤ 1 . As a consequence, the following charge conservation law holds: with M ∈ ℝ . Finally, the potentials V n and V p are assumed to satisfy where the last condition means that the potentials are confining.
The main goal of this paper is to prove exponential convergence to equilibrium of system (1)- (4) with explicit bounds on rates and constants, which are independent of the relaxation time . We therefore consider ∈ (0, 0 ] for arbitrary but fixed 0 > 0 . Our study also includes the limiting case = 0. The main tool in quantifying the large-time behaviour of global solutions to system (1) is the entropy functional For n and p, we encounter contributions of the Boltzmann-entropy form a ln a − (a − 1) ≥ 0 , whereas n tr enters the entropy functional via a non-negative integral term. Note that the integral ∫ n tr 1∕2 ln s 1−s ds is non-negative and well-defined for all n tr (x) ∈ [0, 1] . By introducing the entropy production functional  (6) P ∶= − d dt E, 1 3 Uniform convergence to equilibrium for a family of drift-… it holds (formally) true along solution trajectories of system (1), (2) that The entropy production functional consists of two non-negative flux terms and two equally non-negative reaction terms of the form (a − 1) ln a ≥ 0 . Thus, the entropy E and its production P are non-negative functionals, which formally implies the entropy E to be monotonically decreasing in time.
In a rigorous proof of the entropy decay, one has to control the two reaction terms in (7), which are unbounded for n tr (t, x) → 0, 1 or n(t, x), p(t, x) → 0 . Hence the entropy production is potentially unbounded even for smooth solutions.
In addition, the concentration n tr (t, x) is bounded away from zero and one in the sense that for all times > 0 there exist positive constants = ( 0 , , n , p ) , = (C n , C p , K n , K p ) and a sufficiently small constant ( , C n , C p , K n , K p ) > 0 such that where = 1+ such that the linear and the inverse linear bound intersect at time . As a consequence of (12), there exist positive constants , Γ > 0 (depending on , , , , V n , V p ) such that where 2 2 = Γ 1+ such that the quadratic and the inverse linear bound intersect at the same time . Remark 1.2 (Proof of Theorem 1.1) The existence theory of Theorem 1.1 for the coupled ODE-PDE problem (1) applies standard parabolic methods and pointwise ODE estimates. It relates to previous results like [15] in assuming L ∞ initial data and proving L ∞ -bounds in order to control non-linear terms. The proof is therefore postponed to the Appendix.
Our first main result proves exponential convergence of solutions to (1)-(4) to a unique positive equilibrium state (n ∞ (x), p ∞ (x), n tr,∞ ) , which is stated in detail in Theorem 2.1. Theorem 1.3 (Exponential convergence to equilibrium) Let (n, p, n tr ) be a global weak solution of system (1)-(4) as given in Theorem 1.1 above corresponding to the non-negative initial data (n I , p I , n tr,I ) ∈ L ∞ (Ω) 3 satisfying ‖n tr,I ‖ L ∞ (Ω) ≤ 1 . Then, this solution satisfies the weak entropy production law for all 0 < t 0 ≤ t 1 < ∞ and the following versions of the exponential decay towards the equilibrium: where E I and E ∞ denote the initial entropy and the equilibrium entropy of the system, respectively, and the equilibrium (n ∞ , p ∞ , n tr,∞ ) is given in Theorem 2.1. Moreover, (12) Theorem 1.5 (Entropy-Entropy Production Inequality) Let 0 , n , p , n 0 , p 0 be positive constants and M ∈ ℝ . Let (n ∞ , p ∞ , n tr,∞ ) be the corresponding equilibrium as in Theorem 2.1. Consider an arbitrarily large positive constant M 1 > 0 and non-negative functions (n, p, n tr ) ∈ L 1 (Ω) 3 satisfying the L 1 -bound n, p ≤ M 1 , the L ∞ -bound ‖n tr ‖ L ∞ (Ω) ≤ 1 , and the conservation law Then, there exists an explicitly computable constant C EEP > 0 such that for all ∈ (0, 0 ] the following functional inequality, called entropy-entropy production inequality, holds true: Remark 1. 6 We point out that Theorem 1.5 derives a general functional inequality for admissible functions (n, p, n tr ) , which only share few natural properties like the L 1 -integrability, boundedness of the trapped states and the conservation law with solutions to (1)-(4). It is a nice robustness feature of the entropy method to be based on functional inequalities which can be reused in related contexts, rather than deriving solution-specific estimates. The constant C EEP is independent of ∈ (0, 0 ] . It only depends on the upper bound 0 > 0 , which can be chosen arbitrarily. Remark 1. 7 We emphasise that the EEP inequality (16) does not depend on the lower and upper solution bounds (11)- (13). These bounds are only needed to prove that solutions to (1)-(4) satisfy the weak entropy production law (14), which is neither directly obvious nor part of the existence theory. Therefore, (14) implies that solutions to Theorem 1.1 may only feature singularities of P at time zero due to a lacking regularity of the initial data or due to initial data n tr, The proof of the EEP-inequality of Theorem 1.5 captures in a certain sense the entire non-linear and global dynamics of system (1)-(4). Hence, its derivation is ought to be an involved task. A key step is the proof of a functional EEP-inequality for the special cases of spatially homogeneous concentrations, which fulfil the conservation law (3) and the L 1 -bounds (cf. Proposition 5.3). This core estimate is then extended to the case of arbitrary concentrations satisfying the same assumptions in Proposition 5.5. This extension also forces one to bound √ n − √ n , √ p − √ p , and √ n tr − √ n tr in L 2 (Ω) by the entropy production. Due to the diffusive part in the dynamical equations for n and p, this is easily achieved for the expressions involving n and p by applying Poincaré's inequality (see the Proof of Theorem 1.5 in Sect. 6). However, this is not possible for n tr as no diffusion is acting on n tr . On the other hand, n tr is subject to indirect diffusive effects, which allow for a control on √ n tr − √ n tr in terms of a suitable functional inequality. Indirect diffusive effects occur when a reversible reaction transfers diffusive behaviour from a diffusive species to a non-diffusive species. A first functional inequality which quantifies an indirect diffusion effect was proven in [4] with significant generalisations to reaction-diffusion systems in [8,12], volume-surface reaction-diffusion systems [11] and reaction-diffusion systems with non-linear diffusion [13]. Here, the corresponding estimate is proven in Proposition 5.6 and might also be of independent interest.
Our two last results on system (1)-(4) combine the exponential convergence to equilibrium as proven in Theorem 1.3 with the solution bounds of Theorem 1.1. This entails uniform-in-time solution bounds for n and p as well as exponential convergence to equilibrium in L ∞ (Ω) for n, p, and n tr . As opposed to (15), the convergence result for n tr in Corollary 1.9 holds true without the coefficient .  (17) allow to improve the bounds (12), (13) and to obtain uniform-in-time bounds in the sense that for all > 0 , there exist sufficiently small and -independent constants , , , Γ > 0 such that

3
Uniform convergence to equilibrium for a family of drift-… and for all t ≥ 0 and a.e. x ∈ Ω where t and as well as t 2 ∕2 and Γ intersect at time > 0.

Corollary 1.9
Under the hypotheses of Theorem 1.3, there exist constants The final topic of this paper considers the limit → 0 , which recovers the wellknown Shockley-Read-Hall drift-diffusion-recombination model (see [15,18]): where Remarking that the entropy-entropy production inequality derived in Theorem 1.5 holds uniformly in the fast-reaction parameter 0 < ≤ 0 , one intuitively expects the entropy method and the convergence result of Theorem 1.3 to extend to system (20). Here, we are interested to make this conjecture rigorous also in view of a better general understanding of the equilibration of systems which are derived as fast-reaction limits or quasi-steady-state approximations. One technical point is how to bypass the -dependency of the conservation law (3). The details of this singular limit are subject of the last Sect. 7. Altogether, we prove for system (20) the Theorems 7.3, 7.2 and Corollary 7.5 as corresponding versions of Theorems 1.3, 1.5 and Corollary 1.8.
Up to our knowledge, this is a first result in performing the entropy method in a non-linear reaction-diffusion-type system uniformly in a fast-reaction limit. Note that our approach yields global convergence to equilibrium for all initial data rather than just exponential stability of equilibria as proven, for instance, in a related 1D Poisson-Nernst-Planck system uniformly in the permittivity entering Poisson's equation [17].
The rest of the paper is organised in the following manner. Section 2 proves the existence of a unique equilibrium (Theorem 2.1) as well as uniform-in-bounds of n ∞ , p ∞ and n tr,∞ . In Sect. 3, we collect a couple of technical lemmata, and within Sect. 4, we state a preliminary proposition which serves as a first result towards an EEP-inequality. An abstract version of the EEP-estimate is proven in Sect. 5, first for constant concentrations and based on that also for general concentrations. Section 6 is concerned with the proofs of the EEP-inequality from Theorem 1.5, the announced exponential convergence from Theorem 1.3 and the uniform L ∞ -bounds from Corollary 1.8, whereas Sect. 7 is devoted to the same issues in the situation → 0 . Finally, the proof of Theorem 1.1 is contained in the Appendix.

Properties of the equilibrium
We prove the existence of a unique positive equilibrium (n ∞ , p ∞ , n tr,∞ ) of system (1)-(3) in a suitable (and natural) function space. Note that uniqueness is only satisfied once the total charge M in (3) is fixed. This equilibrium can either be seen as the unique solution of the below stationary system (21) or as the unique state for which the entropy production (7) vanishes. and the conservation law Consequently, the unique positive equilibrium (n ∞ , p ∞ , n tr,∞ ) ∈ X is given by (22)- (24), and Finally, for all M ∈ ℝ and for ∈ (0, 0 ] , there exist two constants ∈ (0, 1∕2) and Γ ∈ (1∕2, ∞) depending only on 0 , n 0 , p 0 , M, V ∶= max(‖V n ‖ L ∞ (Ω) , ‖V p ‖ L ∞ (Ω) ) such that Proof of Theorem 2. 1 We shall prove the equivalence of the statements in the theorem by a circular reasoning. Assume that (n ∞ , p ∞ , n tr,∞ ) ∈ X is a solution of the stationary system (21). In this case, We test Eq. (21a) with ln(n ∞ ∕(n 0 n )) . Due to n ∞ ∈ H 1 (Ω) and n ∞ ≥ a.e. in Ω , the test function ln(n ∞ ∕(n 0 n )) belongs to H 1 (Ω) . We find In the same way, we test Eq. (21b) with ln(p ∞ ∕(p 0 p )) ∈ H 1 (Ω) . This yields Moreover, we multiply (21c) with ln(n tr,∞ ∕(1 − n tr,∞ )) ∈ L 2 (Ω) , integrate over Ω and obtain Taking the sum of the three expressions above, we arrive at (23) n * p * = n 0 p 0 (24) n * n − p * p + n tr,∞ = M.
In order to prove the bounds (26), we observe n tr,∞ = n * n * + n 0 = p 0 p * + p 0 ∈ (0, 1). n * n − p * p + n tr,∞ = M, the left hand side is strictly monotone increasing from −∞ to +∞ as n * ∈ (0, ∞) , while the right hand side is strictly monotone decreasing and bounded between (M, M − 0 ) as n * ∈ (0, ∞) . Both monotonicity and unboundedness/boundedness imply uniform positive lower and upper bounds for n * as explicitly proven in the following: First, we derive that for all ∈ (0, 0 ] . Note that (27) is not an explicit representation of n * since n tr,∞ depends itself on n * . Because of n tr,∞ ∈ (0, 1) , we further observe that where = ( 0 , n 0 , p 0 , M, V) . And as a result of the elementary inequality for a ≥ 0 and b > 0 , we also conclude that where = ( 0 , n 0 , p 0 , M, V) . Similar arguments show that corresponding bounds and are also available for p * . Hence, Due to n ∞ = n * e −V n , p ∞ = p * e −V p and the L ∞ -bounds on V n and V p , the claim of the proposition follows. ◻

Some technical lemmata
A particularly useful relation between the concentrations n, p and n tr is the following Lemma. Proof With n ∞ − p ∞ + n tr,∞ = M (note that n tr,∞ = n tr,∞ is constant), we have p − p ∞ = n − n ∞ + (n tr − n tr,∞ ) . We employ this relation to replace p − p ∞ on the left hand side of (28) and calculate Now, the first term on the right hand side vanishes due to n * p * = n 0 p 0 while we use p * ∕p 0 = (1 − n tr,∞ )∕n tr,∞ for the second term and obtain as claimed above. ◻

Lemma 3.2 (Relative Entropy) The entropy relative to the equilibrium reads
Proof By the definition of E(n, p, n tr ) in (5), we note that n tr,∞ ln s 1 − s ds dx.

3
Uniform convergence to equilibrium for a family of drift-… We expand the first integrand as n ln n n 0 n = n ln n n ∞ + n ln n ∞ n 0 n . Thus, with n ∞ ∕ n = n * , we get Together with an analogous calculation of the p-terms, we obtain ) . This allows us to derive the following Csiszár-Kullback-Pinsker-type inequality:

3
where we applied Hölder's inequality in the last step. ◻ The subsequent lemma provides L 1 -bounds for n and p in terms of the initial entropy of the system and some further constants.

Lemma 3.4 (L 1 -bounds)
Due to the monotonicity of the entropy functional, any entropy producing solution of (1) satisfies Proof Employing Lemma 3.3 and Young's inequality, we find Solving this inequality for n yields Therefore, we arrive at where we used the monotonicity of the entropy functional in the last step. In the same way, we may bound p from above. ◻ At certain points, we will have to estimate the difference between terms like n∕n ∞ and n∕n ∞ . Using Lemma 3.5 below, we can bound this difference by the J nflux term and, hence, by the entropy production.

Lemma 4.2 (Logarithmic Sobolev inequality on bounded domains)
Let Ω be a bounded domain in ℝ m , m ≥ 1 , such that the Poincaré (-Wirtinger) and Sobolev inequalities hold with q = ∞ for m = 1 and any q < ∞ for m = 2 . Then, the logarithmic Sobolev inequality The Log-Sobolev inequality allows to bound an appropriate part of the entropy functional by the flux parts of the entropy production. The normalised variables on the left hand side of the subsequent inequality naturally arise when reformulating the flux terms on the right hand side in such a way that we can apply the Log-Sobolev inequality on Ω.

3
Uniform convergence to equilibrium for a family of drift-… Note that ‖ ‖ L 2 (dx) is in general different from one, whereas ‖ ‖ L 2 (dy) = 1 . We now estimate with ‖V n ‖ L ∞ (Ω) ≤ V and the logarithmic Sobolev inequality (31) The corresponding estimate involving J n reads The same arguments apply to the terms involving p. ◻ The following proposition contains the first step towards an entropy-entropy production inequality. The relative entropy can be controlled by the flux part of the entropy production and three additional terms, which mainly consist of square roots of averaged quantities. The proof that the entropy production also serves as an upper bound for these terms will be the subject of the next section.

Proposition 4.4
There exists an explicit constant C( , Γ, M 1 ) > 0 such that for (n ∞ , p ∞ , n tr,∞ ) ∈ X from Theorem 2.1 and all non-negative functions (n, p, n tr ) ∈ L 1 (Ω) 3 satisfying n tr ≤ 1 , the conservation law and the L 1 -bound the following estimate holds true: (Note that the right hand side of (32) vanishes at the equilibrium (n ∞ , p ∞ , n tr,∞ ).) Proof According to Lemma 3.2, we have Recall that n =ñ n , n ∞ =ñ ∞ n ∞ and ñ ∞ =̃ n . Using these relations, we rewrite the first two integrands as and analogously for the p-terms. This results in The terms in the second line of (33) can be estimated using the Log-Sobolev inequality of Proposition 4.3. Moreover, the elementary inequality x ln x − (x − 1) ≤ (x − 1) 2 for x > 0 gives rise to and an analogous estimate for the corresponding expressions involving p. The second term on the right hand side of the previous line can be bounded from above by applying Lemma 3.5, which guarantees a constant C( , Γ, M 1 ) > 0 such that Uniform convergence to equilibrium for a family of drift-… See (26) and Lemma 3.4 for the bounds on n * , n ∞ and n . We have thus verified that with some c 2 ( , Γ, M 1 ) > 0 . A similar estimate holds true for the corresponding part of (33) involving p.
Considering the last line in (33), we further know that for all x ∈ Ω there exists some mean value such that Consequently, In fact, we will prove that there even exists some constant ∈ (0, 1∕2) such that for all x ∈ Ω . Thus, the function (x) is uniformly bounded away from 0 and 1 on Ω . To see this, we first note that n tr,∞ ∈ [ , 1 − ] using the constant ∈ (0, 1∕2) from (26). In addition, for all x ∈ Ω . Together with (34), this estimate implies We now choose an arbitrary x ∈ Ω and distinguish two cases. If |n tr (x) − n tr,∞ | ≥ ∕2 , then . Again the constant depends only on .
As a result of the calculations above, we may rewrite the last line in (33) as Applying the mean-value theorem to the expression in brackets and observing that we find with a constant c 3 ( ) > 0 after applying the estimate | (x) − n tr,∞ | ≤ |n tr (x) − n tr,∞ | for all x ∈ Ω . Finally, we arrive at (2) .
Uniform convergence to equilibrium for a family of drift-… with a constant C( , Γ, M 1 ) > 0 . ◻

Abstract versions of the EEP-inequality
Notation 5. 1 We set and define the positive constants The motivation for introducing the additional variable n ′ tr is the possibility to symmetrise expressions like (n(1 − n tr ) − n tr ) 2 + (pn tr − (1 − n tr )) 2 as (nn � tr − n tr ) 2 + (pn tr − n � tr ) 2 . Similar terms will appear frequently within the subsequent calculations.
Remark 5. 2 We may consider n ′ tr as a fourth independent variable within our model. In this case, the reaction-diffusion system features the following two independent conservation laws: The special formulation of the first conservation law will become clear when looking at the following two Propositions. There, we derive relations for general variables a, b, c and d, which correspond to √ n∕(n 0 n ) , √ p∕(p 0 p ) , √ n tr and √ n ′ tr , respectively.

3
The following Proposition 5.3 establishes an upper bound for the terms in the second line of (32) in the case of constant concentrations a, b, c and d. This result is then generalised in Proposition 5.5 to non-constant states a, b, c, d.  (25), we also find and Moreover, the two conservation laws from the hypotheses rewrite as

3
Uniform convergence to equilibrium for a family of drift-… The relations (36), (37) allow to express 3 and 4 in terms of 1 and 2 , although not explicitly: where the last definition follows from inserting the previous expression (38) for 3 while the factor 2 + 3 is bounded in Therefore, all the terms f i,j are uniformly positive as well as bounded from above: All constants C i,j and C i,j only depend on 0 , n 0 , p 0 , M, M 1 and V, and there exist corresponding bounds C > 0 and C > 0 such that for all i, j In order to prove (35), we show that under the constraints of the conservation laws (36), (37), respectively, the relations (38), (39), there exists a constant Recall that 2 ∞ ≤ Γ∕n 0 , 2 ∞ ≤ Γ∕p 0 and 2 tr,∞ , � 2 tr,∞ ∈ [ , 1 − ] with ∈ (0, 1∕2) and Γ ∈ (1∕2, ∞) depending on 0 , n 0 , p 0 and M for all ∈ (0, 0 ] (cf. the proof of Proposition 26). Since numerator and denominator of (40) are sums of quadratic terms, it is sufficient to bound the denominator from below in terms of its numerator omitting the prefactors 2 ∞ , 2 ∞ , 2 tr,∞ and � 2 tr,∞ , i.e. to prove that More precisely, we will prove that there exists a constant c( 0 , C, C) > 0 for all ∈ (0, 0 ] such that and that For this reason, we distinguish four cases and we shall frequently use estimates like We mention already here that all subsequent constants c 1 , c 2 are strictly positive and depend only on 0 , C and C uniformly for ∈ (0, 0 ]. As above, ( * ) ≥ c 2 ( 2 1 + 2 2 ) . The signs Uniform convergence to equilibrium for a family of drift-… and, thus, Case 3: 1 < 0 ∧ 2 ≥ 0 : Here, 3 ≥ 0 due to (38) and, thus, 4 ≤ 0 by (39), which yields for all ∈ (0, 0 ] . And as 1 , 4 ≤ 0 ≤ 3 , one has Case 4: 1 < 0 ∧ 2 < 0 : Supposing that 3 ≥ 0 and thus 4 ≤ 0 by (39), we observe Furthermore, 3 ≥ 0 enables us to estimate and Hence, ( * ) . The second estimate in terms of 2 3 follows with 1 , 4 ≤ 0 ≤ 3 from In the opposite case that 3 < 0 and thus 4 ≥ 0 due to (39), we estimate We, thus, arrive at . The corresponding inequality for 3 reads which follows from 2 , 3 ≤ 0 ≤ 4 . The proof of the proposition is now complete. ◻ Notation 5.4 From now on, ‖ ⋅ ‖ without further specification shall always denote the L 2 -norm in Ω.
Within the subsequent Proposition 5.5, the expressions (ad − c) 2 and (bc − d) 2 on the right hand side of (35) will be generalised to ‖ad − c‖ 2 and ‖bc − d‖ 2 in (42). We will later show in the proof of Theorem 1.5 that ‖ad − c‖ 2 (and also ‖bc − d‖ 2 ) can be estimated from above via the reaction terms within the entropy production (7) when using the special choices √ n∕(n 0 n ) , √ p∕(p 0 p ) , √ n tr , and √ n ′ tr for a, b, c, and d.
Uniform convergence to equilibrium for a family of drift-… Proof We divide the proof into two steps. In the first part, we shall derive lower bounds for the reaction terms ‖ad − c‖ 2 + ‖bc − d‖ 2 involving (a d − c) 2 + (b c − d) 2 . This will allow us to apply Proposition 5.3 in the second step.
We now define and split the squares of the L 2 (Ω)-norm as and respectively. In order to estimate the first integral in (43) from below, we write This yields where we used Young's inequality 2xy ≥ −x 2 ∕2 − 2y 2 for x, y ∈ ℝ in the second step and the boundedness of i , 1 ≤ i ≤ 4 , in the last step. Similarly, we deduce The second integral in (43)  Taking the sum of both contributions to (43), we finally arrive at

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Uniform convergence to equilibrium for a family of drift-… and Step 2: We recall that (26) guarantees the uniform positivity and boundedness of ∞ , ∞ , tr,∞ , and � tr,∞ for all ∈ (0, 0 ] in terms of 0 , n 0 , p 0 , M, and V. Therefore, the bounds a 2 , b 2 ≤ C(n 0 , p 0 , M 1 , V) and c 2 , d 2 ≤ 1 give rise to a constant We now want to derive a formula for a in terms of 1 and 1 . Since , one finds and analogous expressions for b , c and d: Furthermore, and, similarly, One observes that the expansions above in terms of 2 i are singular if, e.g., a 2 is zero. We therefore distinguish the following two cases.
Case 1: Uniform convergence to equilibrium for a family of drift-… In order to finish the proof, it is-according to Step 1-sufficient to show that for appropriate constants C 1 , C 2 > 0 . But due to Step 2 it is sufficient to show that for suitable constants C 1 , C 2 > 0, Collecting all 2 i -terms on the right hand side, one only has to prove that or, equivalently, Uniform convergence to equilibrium for a family of drift-… for all ∈ (0, 0 ] . Using an analogue expansion as before, we further deduce with d 2 ≤ 1, As a corresponding estimate holds true also for the other expression on the right hand side of (48), we have shown that there exists a constant C 1 ( , 0 , n 0 , p 0 , M, M 1 , V) > 0 independent of for ∈ (0, 0 ] such that Choosing C 2 > 0 now sufficiently large, Eq. (47) holds true.
Case 2: a 2 < 2 ∨ b 2 < 2 ∨ c 2 < 2 ∨ d 2 < 2 : In this case, we will not need Proposition 5.3 and we shall directly prove (42) employing only the result of Step 1. In fact, for chosen sufficiently small, the states considered in Case 2 are necessarily bounded away from the equilibrium and the following arguments show that consequentially the right hand side of (42) is also bounded away from zero, which allows to close the estimate (42). As a result of the hypotheses a 2 , b 2 ≤ C(n 0 , p 0 , M 1 , V) and c 2 , d 2 ≤ 1 , we use Young's inequality to estimate a, b, c, d ≤ c(n 0 , p 0 , M 1 , V) and with C > 0 uniformly for ∈ (0, 0 ] . We stress that the subsequent cases are not necessarily exclusive.
due to (44) with a constant K( , 0 , n 0 , p 0 , M, M 1 , V) > 0. All arguments within Step 2 remain valid, if we finally set ∶= min( a , b , c , d ) . We also observe that the constants K > 0 above are independent of ∈ (0, 0 ] . And since only depends on n 0 , p 0 , M 1 and V, we may skip the explicit dependence of C 2 on at the end of Case 1. This finishes the proof. ◻ We already pointed out that ‖ad − c‖ 2 and ‖bc − d‖ 2 can be controlled by the reaction terms of the entropy production, if we replace a, b, c, d by √ n∕(n 0 n ) , √ p∕(p 0 p ) , √ n tr and √ n ′ tr (see the proof of Theorem 1.5 in Sect. 6 for details). In this proof, also ‖∇a‖ 2 , ‖∇b‖ 2 , ‖a − a‖ 2 and ‖b − b‖ 2 may be bounded by the entropy production. However, ‖c − c‖ 2 and ‖d − d‖ 2 may not be estimated with the help of Poincaré's inequality since this would yield terms involving ∇n tr , which do not appear in the entropy production.
Instead, we are able to derive the following estimates for ‖c − c‖ 2 and ‖d − d‖ 2 , which describe an indirect diffusion transfer from c to b and from d to a, respectively: Even if c and d are lacking an explicit diffusion term in the dynamical equations, they do experience indirect diffusive effects thanks to the reversible reaction dynamics and the diffusivity of a and b. This is the interpretation of the following functional inequalities. Since c 2 + d 2 = 1 , we obtain

3
Uniform convergence to equilibrium for a family of drift-… where we applied (50) in the last step. Consequently, and using (49). ◻

EEP-inequality and convergence to the equilibrium
We are now prepared to prove Theorem 1.5.
Proof of Theorem 1.5 Let (n, p, n tr ) ∈ L 1 (Ω) 3 be non-negative functions satisfying n tr ≤ 1 , the conservation law n − p + n tr = M and the L 1 -bound n, p ≤ M 1 . Keeping in mind that ∞ = √ n * ∕n 0 and ∞ = √ p * ∕p 0 (cf. Notation 5.1), Proposition 4.4 guarantees that there exists a positive constant C 1 ( , Γ, M 1 ) > 0 such that Next, we have to bound the second line of (51) in terms of the entropy production. To this end, we apply Proposition 5.5 with the choices a ∶= √ n∕(n 0 n ) , b ∶= √ p∕(p 0 p ) , c ∶= √ n tr and d ∶= √ n � tr (as always n � tr = 1 − n tr ). The hypotheses of this proposition are fulfilled as a consequence of the conservation law n − p + n tr = M and the L 1 -bound n, p ≤ M 1 . As a result, we obtain for all ∈ (0, 0 ] with a constant C 2 ( 0 , n 0 , p 0 , M, M 1 , V) > 0 . Thanks to Poincaré's inequality, we are able to bound the second and third line on the right hand side from above: and Moreover, the elementary inequality ( √ x − 1) 2 ≤ (x − 1) ln(x) for x > 0 gives rise to and similarly for a constant C 4 ( 0 , n , p , n 0 , p 0 , M, M 1 , V) > 0 independent of ∈ (0, 0 ] . ◻ Theorem 1.5 provides an upper bound for the relative entropy in terms of the entropy production. This already implies exponential convergence of the relative entropy. The subsequent proposition now yields a lower bound for the relative entropy involving the L 1 -distance of the solution to the equilibrium. This will allow us to establish exponential convergence in L 1 .  ∈ (0, 1) . Thus, there exists some (s) between n tr,∞ and s such that E(n, p, n tr ) − E(n ∞ , p ∞ , n tr,∞ ) � . Now, we are able to prove exponential convergence in relative entropy and L 1 .

E(n, p, n tr
Proof of Theorem 1. 3 We first prove exponential convergence of the relative entropy using a Gronwall argument as stated in [25]. To this end, we choose 0 < t 0 ≤ t 1 ≤ t < T and rewrite the entropy production law as where we applied Theorem 1.5 with K ∶= C −1 EEP in the second step. Furthermore, we set and obtain from (52) the estimate KΨ(t 1 ) ≤ (t 1 ) − (t) which yields Integrating this inequality from t 1 = t 0 to t 1 = t and observing that Ψ(t) = 0 gives rise to As a consequence of (52) with t 1 = t 0 , one has −Ψ(t 0 ) ≥ ( (t) − (t 0 ))∕K and, hence, But this is equivalent to for all t ≥ t 0 > 0 . In order to conclude that for all t ≥ 0 , we observe that the rate K is independent of t 0 and that the entropy E(n, p, n tr )(t 0 ) extends in (53) continuously to t 0 → 0 since n, p ∈ C([0, T);L 2 (Ω)) for all T > 0 by Theorem 1.1. This results in the announced exponential decay of the relative entropy, while the exponential convergence in L 1 follows from Proposition 6.1. ◻ Proof of Corollary 1. 8 We first prove that the linearly growing L ∞ -bounds together with parabolic regularity for system (1) and assumption (4) entail polynomially growing W 1,q -bounds, q ∈ (1, ∞) , for n and p. To this end, we consider and introduce the variable w = n e V n . We observe that ∇ ⋅ J n = ∇ ⋅ e −V n ∇w = e −V n Δw − ∇V n ⋅ ∇w and thus, Under the assumptions of Corollary 1.8, this equation is of the form (53) E(n, p, n tr )(t) − E(n ∞ , p ∞ , n tr,∞ ) ≤ (E(n, p, n tr )

3
Uniform convergence to equilibrium for a family of drift-… Using the inequalities |ab| ≤ (a 2 + b 2 )∕2 and (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 ) for a, b, c ∈ ℝ , we find x ∈ Ω , we derive An integration by parts and Young's inequality with C 1 (C, q) > 0 give rise to Hence, there exists a constant C 2 (C, q) > 0 such that where A, B > 0 result from the linearly growing L ∞ -bounds from (11). For any fixed t 0 > 0 and all t ≥ t 0 , we now have A Gronwall lemma (see e.g. [2]) now proves the desired polynomial growth of ‖∇w‖ L q (Ω) and ‖∇n‖ L q (Ω) : Next, we use the Gagliardo-Nirenberg-Moser interpolation inequality in ℝ m , m ≥ 1 (see e.g. [23]): Then, interpolating with the exponentially decaying L 1 -norm of n − n ∞ , we obtain due to the exponential convergence to equilibrium (15). The estimate for p follows in the same way. ◻ Proof of Corollary 1. 9 We first notice that exponential convergence of n and p in L q (Ω) , 1 < q < ∞ , immediately follows from Theorem 1.3 and Corollary 1.8 via and an analogous estimate for p where 0 < C q , K q < ∞ are constants independent of ∈ (0, 0 ] . Reusing the Gagliardo-Nirenberg-Moser interpolation inequality in ℝ m , m ≥ 1 , from (56) and the polynomial bound on the growth of ‖∇n(t)‖ L 2m (Ω) from (55), we derive thus, establishing exponential convergence of n and p in L ∞ (Ω) . Concerning the convergence of n tr , we recall the following identities from (25) and (22): We abbreviate u∶=n tr − n tr,∞ and calculate (pointwise in x) by adding and subtracting n tr,∞ and n tr multiple times and by using the relations (58): Moreover, there exist positive constants C n (‖n I ‖ L ∞ (Ω) , V n ) , C p (‖p I ‖ L ∞ (Ω) , V p ) and K n (V n ) , K p (V p ) such that Finally, there exist positive constants , Γ , > 0 (depending on , C n , C p , K n , K p , V n , V p ) such that where = Γ 1+ such that the bounds t and Γ∕(1 + t) intersect at time .
The entropy functional (5) extends continuously to the limit = 0: which is again an entropy (the free energy) functional of the Shockley-Read-Hall model (20). The corresponding entropy production (free energy dissipation) functional reads as Next, we recall from the introduction n qssa tr = n qssa tr (n, p) such that R n (n, n qssa tr ) = R p (p, n qssa tr ) , i.e. n qssa tr (n, p) denotes the pointwise equilibrium value of the trapped states in (1) for fixed n and p, which corresponds to the quasi-steady-state approximation = 0.
Moreover, we observe that the Shockley-Read-Hall entropy production functional (61) can be identified as the entropy production functional P(n, p, n qssa tr ) as given in (7) along trajectories of (1) with = 0 when n tr ≡ n qssa tr (n, p): for all t ≥ 0.
We are now in the position to formulate the EEP-inequality. Then, the following EEP-inequality holds true for all non-negative functions (n, p) ∈ L 1 (Ω) 2 satisfying the conservation law n − p = M, the L 1 -bound n, p < M 1 as well as the conditions E 0 (n, p) < ∞ , P 0 (n, p) < ∞ , P(n, p, n qssa tr ) < ∞ for some 0 > 0: where C EEP > 0 is the same constant as in Theorem 1.5. Theorem 7.3 (Exponential convergence for = 0 ) Let (n, p) be a global weak solution of system (20) as given in Theorem 7.1 corresponding to the non-negative initial data (n I , p I ) ∈ L ∞ (Ω) 2 . Then, this solution satisfies the entropy production law Moreover, the following versions of the exponential decay towards the equilibrium (n ∞,0 , p ∞,0 ) ∈ X 0 hold true: where C ∶= C −1 CKP and K ∶= C −1 EEP are the same constants as in Theorem 1.3. Moreover, E I and E ∞ denote the initial entropy of the system and the entropy in the equilibrium, respectively. Remark 7. 4 We believe that the entropy-entropy production inequality (64) can also be proven by combining estimates of Sect. 5 with previous works on the entropy method for detailed balanced reaction-diffusion models, see e.g. [5,7,10,19]. We emphasise, however, that our goal with Theorem 7.2 is to be able to derive an entropy-entropy production inequality via the fast-reaction parameter → 0.
Finally, in the same way as for strictly positive > 0 , we can derive uniformin-time L ∞ -bounds for n and p also in the case = 0 . As before, these bounds further improve the lower bounds on n and p. In order to resolve this issue, we shall apply the EEP-inequality from Theorem 1.5 to a suitably defined sequence of functions (n , p , n tr, ) ∈ L 1 (Ω) 3 which fulfil ‖n tr, ‖ L ∞ (Ω) ≤ 1 , the L 1 -bound n , p ≤ M 1 and the conservation law  (62). For this choice, we derive the stated EEP-estimate for the case = 0 via the following steps, which are proven below: We recall that n and p are assumed to satisfy E 0 (n, p) < ∞ and P 0 (n, p), P(n, p, n qssa tr ) < ∞ , which implies that P 0 (n, p) = P(n, p, n qssa tr ) as discussed in the introduction.
Step 1. Proof of (69): We first show, that with (n , p , n tr, ) = (n, p , n qssa tr ) Recalling that we first notice that p = p + n qssa tr → p monotonically decreasing for → 0 for all x ∈ Ω . Thus, by using n qssa tr ≤ 1 and the elementary estimate p ln p ≤ 2p (ln p + ln 2) for p ≥ max{ 0 , 1} , the Lebesgue dominated convergence theorem, the L 1 -bounds n, n , p, p ≤ M 1 and E 0 (n, p) < ∞ imply the convergence of the p -integral in (72). The convergence of the third integral follows directly from Using analogue arguments, the convergence follows from observing the monotone convergence n * → n * ,0 and p * → p * ,0 for → 0 due to (27) in the proof of Theorem 2.1, which directly implies the monotone convergence n ∞ → n ∞,0 and p ∞ → p ∞,0 for all x ∈ Ω , where (n ∞ , p ∞ , n tr,∞ ) and (n ∞,0 , p ∞,0 ) are defined in (22) and (63), respectively. n − p + n tr, = M. where C EEP > 0 is the same constant as in Theorem 1.5.
Step 3. Proof of (71): As the constant C EEP > 0 is independent of ∈ (0, 0 ] , it suffices to show that To this end, we consider the representation where we have already taken into account that n = n , ∇p = ∇p and n tr, = n qssa tr for all > 0. We note first that the convergence of the second, third and forth integral follows from the pointwise convergence of p for all x ∈ Ω and from the Lebesgue dominated convergence theorem by estimating where the function on the right hand side is integrable due to the finiteness of P(n, p, n qssa tr ). Secondly, the product n − p + n tr, = n − p = M E(n , p , n tr, ) − E(n ∞ , p ∞ , n tr,∞ ) ≤ C EEP P(n , p , n tr, ) lim →0 P(n , p , n tr, ) = P(n, p, n qssa tr ).  converges pointwise for all x ∈ Ω as → 0 . In order to conclude the convergence of the corresponding integral via the Lebesgue dominated convergence theorem, we use similar to Step 1 the elementary inequality p ln p ≤ 2p (ln p + ln 2) for p ≥ max{ 0 , 1} and the finiteness of P(n, p, n qssa tr ) . This yields and therefore, P(n , p , n tr, ) → P(n, p, n qssa tr ) for → 0 . ◻ Proof of Theorem 1. 3 We only have to check that the assumptions on the finiteness of the entropy E 0 and the entropy production functionals P 0 and P within Theorem 2 are satisfied. The claim of this theorem then follows from the same arguments as in the proof of Theorem 1.3. Due to the uniform L ∞ -bounds (59) of n(t) and p(t) for all t ≥ 0 , we know that E 0 (n, p) < ∞ for all t ≥ 0 . Similarly, we deduce that P(n, p, n qssa tr ) and P 0 (n, p) are finite for all strictly positive t > 0 since n, p are bounded away from zero and n qssa tr is bounded away from zero and one uniformly in Ω.
Finally, the lower bounds (60) guarantee similar to Theorem 1.5 that solutions satisfy the weak entropy production law (65) for all t 0 > 0 . ◻

Conclusion
We have investigated the drift-diffusion-recombination system (1) modelling the transport, generation and annihilation of negatively charged electrons and positively charged holes (vacancies of electrons) in certain types of semiconductors. As depicted in Fig. 1, we have considered a two-level system augmented by an additional intermediate energy level, the so-called trap level, which results from the presence of foreign atoms inside the crystal of the semiconductor. We have derived an entropy-entropy production (EEP) inequality (cf. Theorem 1.5) which bounds the entropy functional (5) from above in terms of the entropy production functional (7). This EEP-inequality has then be used to show that the concentrations of electrons and holes converge to their equilibrium distributions at an exponential rate as time tends to infinity (cf. Theorem 1.3). A novel achievement of our studies is the fact that the entropy method has been applied uniformly in a small time-related parameter. More precisely, the constant C EEP in Theorem 1.5 is independent of the lifetime of electrons on the trap level (cf. (1)) provided ∈ (0, 0 ] for some 0 > 0 . The -independence of C EEP transfers to the constants appearing in the exponential decay estimate in Theorem 1.3. This proves that the exponential convergence rate is independent of a quasi-steady-state dx approximation of the electrons on the trap level, which leads to the famous Shockley-Read-Hall recombination model [16,21]. In particular, we were able to derive an EEP-inequality and the convergence estimate for the limiting Shockley-Read-Hall model. This fact is notable from a conceptual point of view as we transfer the results for > 0 to the case = 0 by performing the limit → 0 . We believe that our limiting approach to the Shockley-Read-Hall model may serve as an example for possible applications of this technique to fast-reaction limits and quasi-steady-state approximations. In view of the technicalities of the proofs and the resulting length of the current paper, our results are still limited by not taking into account the self-consistent potential generated by electrons and holes, which is required by a physically more precise model. However, this leads to an additional coupling of (1) to Poisson's equation and a further increase in complexity of the problem. We expect however to resolve these issues in a future work by combining techniques and results presented in the current paper with ideas in [9], which considered a self-consistent Shockley-Read-Hall model without trapped states.

3
Uniform convergence to equilibrium for a family of drift-… One obtains which results in Analogously, we derive For convenience, we also introduce the abbreviations as well as , > 0 such that the following estimates hold true a.e. in Ω: Next, we introduce the new variable for reasons of symmetry. In fact, we can prove the positivity of n ′ tr in the same way as for n tr , which then implies the desired bound 0 ≤ n tr ≤ 1 . A further ingredient for establishing the positivity of the variables u, v, n tr and n ′ tr is to project them onto [0, ∞) and [0, 1], respectively, on the right hand side of the PDE-system. In this context, we use X + ∶= max(X, 0) to denote the positive part of an arbitrary function X and X [0,1] ∶= min(max(X, 0), 1) for the projection of X to the interval [0, 1].  The no-flux boundary conditions of (1) transfer to similar conditions on u and v. In detail, we have and, hence, Therefore, the corresponding boundary conditions for u and v read Furthermore, we assume that the corresponding initial states satisfy In this situation, ‖n tr,I ‖ L ∞ (Ω) + ‖n � tr,I ‖ L ∞ (Ω) ≥ 1 and we set We now aim to apply Banach's fixed-point theorem to obtain a solution of (74)-(76).
Step 1: Definition of the fixed-point iteration. For any time T > 0 (to be chosen sufficiently small in the course of the fixed-point argument), we introduce the space and the closed subspace The fixed-point mapping S ∶ X T → X T is now defined via where (u, v, n tr , n � tr ) is the solution of the following PDE-system subject to the boundary and initial conditions specified above: (75) n ⋅ ∇u + 1 2 u∇V n =n ⋅ ∇v + 1 2 v∇V p = 0.
(76) (u I , v I , n tr,I , n � tr,I ) ∈ L ∞ + (Ω) 4 , n tr,I + n � tr,I = 1. S(ũ,ṽ,ñ tr ,ñ � tr ) ∶= (u, v, n tr , n � tr ) This inequality already implies a linear bound on the L ∞ -norm of u as we shall see below (cf. [2]). We define and note that U(0) = 0 . Estimate (78) entails for all t ∈ [0, T] , where > 0 is an arbitrary constant, which guarantees that the expression X ∶= + ‖u(0)‖ q L q (Ω) + U(t) is strictly positive. Multiplying both sides with X (1−q)∕q and integrating from 0 to t gives We now substitute ∶= U(s) and deduce where we have used (78) in the last step. Therefore, and, taking the limit → 0, As the bound on the right hand side is independent of q, we even obtain for all t ∈ [0, T] . This result naturally gives rise to An analogous estimate is valid for v. As a result, we obtain for T > 0 chosen sufficiently small.
Employing (79), we also derive The same argument is applicable to v, which results in for sufficiently small T > 0 . The corresponding bounds on n tr and n ′ tr can be deduced from the formula and from an analogous one for n ′ tr . In fact, and, hence, for T > 0 sufficiently small.

3
Uniform convergence to equilibrium for a family of drift-… In a similar way, we arrive at Due to n tr (0) = 0 , one obtains for t ∈ [0, T] and, using similar techniques as above, Note that because of f 4 = −f 3 , the last estimate equally serves as an upper bound for ‖n � tr (t)‖ L 2 (Ω) . Taking the sum of the above estimates and choosing T > 0 sufficiently small yields with some c ∈ (0, 1).
Step 2 and Step 3 imply that for T > 0 sufficiently small the mapping S ∶ M T → M T is a contraction. Banach's fixed point theorem, thus, guarantees that there exists a unique (u, v, n tr , n � tr ) ∈ M T such that S(u, v, n tr , n � tr ) = (u, v, n tr , n � tr ) . Moreover, due to standard parabolic regularity for (u, v), the fixed-point (u, v, n tr , n � tr ) is the unique weak solution of ‖(u, v, n tr , n � tr )‖ X T ≤ c ‖(ũ,ṽ,ñ tr ,ñ � tr )‖ X T In order to prove the non-negativity of u, v, n tr and n ′ tr , we adapt an argument from [26]. First, we define on [0, T] × Ω and notice that h ≤ 0 and h(t = 0) = 0 a.e. since u(0) ≥ 0 a.e. We now multiply the first equation in (81) with h and integrate over (0, t) × Ω for t ∈ [0, T] . This yields The first term on the right hand side of (82) can be seen to be non-positive using integration by parts and the boundary condition from (75): due to uh ≥ 0 , n ⋅ ∇V n ≥ 0 , and since ∇h ≠ 0 holds true only in the case u < 0 , where we have ∇u = ∇h in L 2 , see e.g. [14]. Moreover, and the third term in (82) is again non-positive as an integral over non-positive quantities: as a consequence of u + h = 0 in L 2 (Ω) . The left hand side of (82) can be reformulated as For the first step, we have used that the integrand s u h only contributes to the integral if h < 0 . But in this case, u = h and, hence, s u = s h in L 2 , see e.g. [14]. This
The non-negativity of n tr follows from a similar idea using Again, h ≤ 0 and h(t = 0) = 0 due to n tr (0) ≥ 0 . Multiplying the third equation of (81) with h and integrating over (0, t) × Ω , t ∈ [0, T] , we find As before, all terms under the integral on the right hand side involving n [0,1] tr vanish. Consequently, for all t ∈ [0, T] . The same result holds true for n ′ tr . Therefore, we have verified that n tr (t, x) , n � tr (t, x) ≥ 0 for all t ∈ [0, T] and a.e. x ∈ Ω. The non-negativity of n tr and n ′ tr together with n � tr = 1 − n tr from (73) now even imply This allows us to identify the unique weak solution (u, v, n tr , n � tr ) of (81) to equally solve which is the transform version of the original problem (1).
Up to now, we have proven that there exists a unique solution (u, v, n tr ) ∈ C([0, T], L 2 (Ω)) 3 such that (u, v, n tr , 1 − n tr ) ∈ M T on a sufficiently small time interval [0, T].
Step 6: L ∞ -bounds for n and p. We now prove the linearly growing L ∞ -bounds (11) for n and p. We only detail the bound for p and sketch how the bound for n follows in a similar fashion. After recalling (with p = 1 w.l.o.g.)