Quasi-Steady-State Approximation: Analysis of a System with Nonnegative Functions, Summaries of Chemistry

Download Quasi-Steady-State Approximation: Analysis of a System with Nonnegative Functions and more Summaries Chemistry in PDF only on Docsity! Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 2 (2007), No. 4, pp. 823-850 QUASI–STEADY–STATE APPROXIMATION FOR REACTION–DIFFUSION EQUATIONS BY MARZIA BISI, FIAMMETTA CONFORTO AND LAURENT DESVILLETTES Abstract In this paper, we present a rigorous proof of the quasi– steady–state approximation (QSSA) used in chemistry, in two different settings: the first one corresponds to reaction–diffusion equations, while the second one is devoted to ODEs, with a par- ticular attention to the effect of temperature. 1. Introduction 1.1. Quasi–steady–state approximation The quasi–steady–state approximation (denoted from now on by QSSA) is a standard procedure in the study of chemical reactions kinetics in situa- tions where certain species have a very short time of existence (free radicals, very unstable molecules, etc.) with respect to other species. It consists in assuming that the variation of the unstable species is zero, so that the size of the set of equations modeling the reactions is reduced (by the number of unstable species). We refer to [17] for a detailed description of the QSSA and the assumptions underlying its validity. When the modeling of the chemistry is done through ODEs, and the unknown is the concentration of species, the rigorous proof of the validity of the QSSA is a consequence of standard theorems of singular perturbation theory. In this frame, QSSA has been studied in a lot of papers in order Received March 9, 2007. AMS Subject Classification: 34E15, 35K57, 80A30, 92E20. Key words and phrases: Quasi-steady-state approximation, reaction-diffusion. 823 824 M. BISI, F. CONFORTO AND L. DESVILLETTES [December to build up reduction algorithms for systems appearing in chemistry (see for instance [1, 6, 16, 17] and the references therein). We also refer to [14], [2] and the references therein for a description of a related procedure in the context of ODEs, namely the fast reaction limit. Our paper is dedicated to the proof of the validity of the QSSA when the unknowns which model the chemistry are more complicated than the time-dependent concentrations of the species. We focus on two such situa- tions: the first one concerns a case in which the unknown is the time-space dependent concentration of the species (and ODEs are replaced by reaction- diffusion PDEs), while the second one deals with the coupling of ODEs for the concentration of the species and the temperature of the mixture. Though those two situations are quite different, we present a mathematical analysis which is based on the same concept: namely, the use of a priori estimates based on the entropy and entropy dissipation. This restricts our result to the cases when the chemistry is reversible. We intend, in the first part of this paper, to show that the QSSA can also be rigorously established in the case when the spatial structure of the mixture is taken into account. We restrict ourselves here to the simplest possible modeling of this spatial structure: namely, when the evolution of the species is made through reaction–diffusion PDEs. We keep however the possibility of having different diffusion rates for different species, and we look for global solutions with general data in any dimension. This problem has been investigated in [18] in cases when there is a bounded invariant region for the unknowns and, when it is not the case, for times smaller than a critical time (depending on the initial data). The methods used in [18] rely on energy bounds for parabolic PDEs, whereas our approach is rather based on Lyapounov functionals, and do not use bounded invariant regions. It is however restricted to reversible chemistry. A related problem consists in looking to the fast reaction limit in reaction- diffusion PDEs. This has been performed both for irreversible reactions [12] and more recently also for reversible ones [3, 4]. This last paper is closest to our approach, since it uses heavily Lyapounov functionals techniques. We wish therefore to emphasize the differences with our work : First, we work in a situation in which no bounded invariant domain is available (because we consider a system of 5 equations), whereas the system of two equations appearing in [4] admits such a domain. As a consequence, we have to rely on L2 estimates obtained by an entropy method or a duality method (cf. [9]). 2007] QUASI–STEADY–STATE APPROXIMATION 827 ∂tb ε − d2 ∆xb ε = 1 ε mε − aεbε , (5) ∂tc ε − d3 ∆xc ε = 1 ε mε − cεdε , (6) ∂td ε − d4 ∆xd ε = 1 ε mε − cεdε , (7) ∂tm ε − d5 ∆xm ε = aεbε + cεdε − 2 ε mε . (8) The initial and boundary conditions write aεi (0, x) = ai0(x) ≥ 0 , i = 1, . . . , 4, mε(0, x) = 0 , x ∈ Ω , (9) n(x) · ∇xa ε i (t, x) = 0, i = 1, . . . , 4, n(x) · ∇xm ε(t, x) = 0 , x ∈ ∂Ω. (10) The formal computation corresponding to the QSSA theory is the fol- lowing: when ε goes to 0, we replace the left-hand side of eq. (8) by 0, and we use the corresponding relation a b+ c d = 2 ε m in eqs. (4)-(7). This leads formally to the following system: ∂ta− d1 ∆xa = 1 2 (c d − a b) , (11) ∂tb− d2 ∆xb = 1 2 (c d − a b) , (12) ∂tc− d3 ∆xc = −1 2 (c d− a b) , (13) ∂td− d4 ∆xd = −1 2 (c d − a b) , (14) with Neumann boundary conditions and initial data: ai(0, x) = ai0(x) ≥ 0 , i = 1, . . . , 4, x ∈ Ω , (15) n(x) · ∇xai(t, x) = 0 , i = 1, . . . , 4, x ∈ ∂Ω . (16) We give a rigorous result corresponding to the formal computation above. It constitutes our first main theorem : Theorem 1.1. Let N ≥ 1, Ω be a bounded regular open set of R N , and ai0, i = 1, . . . , 4, be nonnegative functions from Ω to R satisfying ∫ Ω |ai0|2 (1 + | ln ai0|2)dx < +∞. Let di, i = 1, . . . , 5 be strictly positive diffusion rates. Then, 828 M. BISI, F. CONFORTO AND L. DESVILLETTES [December 1. For any ε > 0, there exists a weak solution aε, bε, cε, dε, mε in ( L2 loc ( [0, +∞[ ; L2(Ω) ) )5 to system (4)-(10). 2. When ε→ 0, there exists a subsequence of aε, bε, cε, dε,mε (still denoted by aε, bε, cε, dε,mε) which converges to a, b, c, d, 0 in ( L1 loc ( [0,+∞[ ; L1 (Ω) ) )5 . Moreover, this limit is a weak solution of system (11)-(16) belonging to ( L2 loc ( [0,+∞[ ; L2(Ω) ) )4 . 3. When N = 1, the solution of (4)-(10) and (11)-(16) is strong and unique as soon as the initial data ai0, i = 1, . . . , 4 are smooth (C2(Ω̄)) and com- patible with the Neumann boundary conditions. Then, the whole sequence aε, bε, cε, dε,mε converges to a, b, c, d, 0. The proof of this theorem is based on a priori estimates mainly coming out of the entropy and entropy dissipation related to system (4)-(10). In this respect, it is reminiscent of the papers [4, 7, 8]. When N > 1, it also uses the method of duality proposed by M. Pierre (cf. [9, 15]) in order to prove an “L2 (lnL)2” bound. 1.3. Presentation of the main result concerning ODEs We introduce again the simple chemical mechanism treated in the first part of our work. Precisely, we consider a mixture of four gases A, B, C, D undergoing a reversible bimolecular chemical reaction through an unstable state, that is A+B ⇄M, M ⇄ C +D, (17) where M is an unstable species. We recall that the fact that this mechanism is reversible plays a decisive role in our analysis. The number densities of species A, B, C, D and M are denoted, respec- tively, by nA, nB, nC , nD and nM . Moreover, the total number density is defined as n = nA + nB + nC + nD + nM . Finally, EA, EB, EC , ED and EM denote the (constant) chemical bond energies of species A, B, C, D and M (they can all be supposed to be nonnegative), and we introduce the notations Eα = EA + EB − EM , Eβ = EC + ED − EM , (18) 2007] QUASI–STEADY–STATE APPROXIMATION 829 for the strength of reactions (17), respectively, and ∆E = Eβ − Eα = EC + ED −EA − EB. (19) Here we assume ∆E ≥ 0 (the other case being similar, since the species A, B, C, D can be ordered). The chemical energy of the mixture is defined as ech = EAnA + EBnB + ECnC + EDnD + EMnM . (20) We assume that the state equations for pressure and energy are those of monoatomic perfect gases (though other laws could be treated in the same way). The total energy of the system, which takes into account the thermal and chemical contributions, is therefore etot = 3 2 nΘ+ ech , (21) where Θ is the temperature of the mixture. The evolution of the mixture is governed by the following set of ODEs [10]: n′A = n′B = −ϕ(Θ)S1, (22) n′C = n′D = −ψ(Θ)S2, (23) n′M = ϕ(Θ)S1 + ψ(Θ)S2, (24) e′tot = 0, (25) where S1 = nA nB − nM e−Eα/Θ+ 3 2 lnΘ− 5 2 , (26) S2 = nC nD − nM e−Eβ/Θ+ 3 2 lnΘ− 5 2 , (27) and ϕ, ψ are given by a heuristic formula such as the one proposed in [10]: ϕ(Θ) = A1Θ B1 exp ( −E1 Θ ) , ψ(Θ) = A2Θ B2 exp ( −E2 Θ ) , (28) where A1,A2, E1, E2 > 0 and B1,B2 ∈ R. Eq. (25) can be written under the explicit form ( EAnA + EBnB + ECnC + EDnD + EMnM + 3 2 nΘ )′ = 0 , (29) 832 M. BISI, F. CONFORTO AND L. DESVILLETTES [December where n = nA + nB + nC + nD and F = F (nA, nB , nC , nD,Θ) ≡ ϕ(Θ)ψ(Θ) ϕ(Θ) + ψ(Θ)e−∆E/Θ ( nCnD − nAnBe −∆E/Θ ) , together with the initial conditions nA(0) = nA0 > 0 , nB(0) = nB0 > 0 , nC(0) = nC0 ≥ 0 , nD(0) = nD0 ≥ 0 , Θ(0) = Θ0 > 0 . (46) We now present our second main theorem, which makes rigorous this formal asymptotics: Theorem 1.2. We assume that ε ∈]0, 1[, that ϕ,ψ are C1 functions of the temperature such that Θ > 0 =⇒ ϕ(Θ), ψ(Θ) > 0 (this assumption includes formulas such as (28)). We consider energies EA, EB , EC , ED > 0 and initial data nA0, nB0 > 0, nC0, nD0 ≥ 0, Θ0 > 0. Then, 1. There exists a unique solution nεA, n ε B, n ε C , n ε D, n ε M , Θε in C1([0,+∞[) to the Cauchy problem (33)-(38). 2. For all t > 0, nεA(t), n ε B(t), n ε C(t), n ε D(t), n ε M (t),Θε(t) > 0. (47) 3. There exist constants c1, c2, . . . > 0 (depending only on the data and independent of ε) such that for all t > 0, nεA(t), n ε B(t), n ε C(t), n ε D(t), n ε M (t) ≤ c1, (48) c2 ≤ nε(t) ≤ c3, (49) c4 ≤ Θε(t) ≤ c5. (50) 4. There exists a unique solution nA, nB, nC , nD, Θ in C1([0,+∞[) to the Cauchy problem (43)-(46). 5. For all t > 0, nA(t), nB(t), nC(t), nD(t),Θ(t) > 0. (51) 6. For the same constants c1, c2, . . . > 0 as in point 3, and for all t > 0, nA(t), nB(t), nC(t), nD(t) ≤ c1, (52) c2 ≤ n(t) ≤ c3, (53) c4 ≤ Θ(t) ≤ c5. (54) 2007] QUASI–STEADY–STATE APPROXIMATION 833 7. There exist constants c6, c7, c8, c9 > 0 (depending on the data and inde- pendent of ε) such that for any ε ∈ ]0, 1[ and T > 0, sup t∈[0,+∞[ nεM (t) ≤ c6 e − 1 εΘε ≤ c6 e − 1 c4 ε , (55) sup t∈[0,T ] ∣ ∣nεA,B,C,D(t)− nA,B, C,D(t) ∣ ∣ ≤ c7 e c8 T ε−1 e − 1 c4 ε , (56) sup t∈[0,T ] |Θε(t)−Θ(t)| ≤ c9 e c8 T ε−1 e − 1 c4 ε . (57) The proof of this theorem is based on the entropy and entropy dissi- pation estimates, which allow to bound from below the temperature of the mixture, and on an estimate showing that the concentration nεM(t) of the un- stable species is bounded by a constant times exp(−1/(εΘε(t))). Note that standard theorems of singular perturbation for ODEs cannot be applied a priori because of the singularities in the data (this problem disappears once it is shown that the temperature is bounded below) and because of the in- terplay of different scales (this is apparent for example in formulas (35)-(37), where ε−1 and e− 1 εΘε appear). 2. Proof of Theorem 1.1 For the basic functional analysis results involved in the sequel, the reader is referred to [5]. We begin by recalling that for a given ε > 0, the existence of a weak solution aε, bε, cε, dε,mε in ( L2 loc ( [0,+∞[ ; L2(Ω) ) )5 to system (4)-(10) is a direct consequence of the results of [9], based on the duality method de- scribed in [15]. 2.1. A priori estimates We begin with the obvious partial conservation of concentrations: Lemma 2.1. The solutions of (4)-(10) conserve the quantities M13 ≡ ∫ Ω (aε(t, x) + cε(t, x) +mε(t, x)) dx = ∫ Ω (a0(x) + c0(x)) dx , (58) M14 ≡ ∫ Ω (aε(t, x) + dε(t, x) +mε(t, x)) dx = ∫ Ω (a0(x) + d0(x)) dx , (59) 834 M. BISI, F. CONFORTO AND L. DESVILLETTES [December M23 ≡ ∫ Ω (bε(t, x) + cε(t, x) +mε(t, x)) dx = ∫ Ω (b0(x) + c0(x)) dx . (60) Proof. The proof follows immediately by integrating over Ω suitable linear combinations of eqs. (4)-(8).  Next we turn to the consequence of the entropy structure of the chemical reactions (this structure is directly related to the reversible character of the problem under consideration). We denote by C any constant, by CT any constant depending on T , etc. Lemma 2.2. Let aεi , i = 1, . . . , 4, and mε be solutions of the system (4)-(10), with initial data ai0 such that ai0 ln(ai0)∈L1(Ω). Then, for all T >0 i = 1, . . . , 4, ‖∇x √ aεi‖2L2([0,T ]×Ω) ≤ CT , (61) ‖∇x √ mε‖2L2([0,T ]×Ω) ≤ CT , (62) i = 1, . . . , 4, sup t∈[0,T ] ‖aεi ln aεi‖L1(Ω) ≤ CT , (63) sup t∈[0,T ] ‖mε ln(mε/ε)‖L1(Ω) ≤ CT , (64) ∫ T 0 ∫ Ω (aεbε −mε/ε) (ln(aεbε)− ln(mε/ε)) dxdt ≤ CT , (65) ∫ T 0 ∫ Ω (cεdε −mε/ε) (ln(cεdε)− ln(mε/ε)) dxdt ≤ CT . (66) Proof. Calculating the time-derivative of the entropy-functional, E, E (t) ≡ ∫ Ω 4 ∑ i=1 (aεi ln a ε i − aεi ) dx+ ∫ Ω (mε ln(mε/ε)−mε) dx (67) yields ∫ Ω 4 ∑ i=1 (aεi ln a ε i − aεi )(T ) dx+ ∫ Ω (mε ln(mε/ε) −mε)(T ) dx + 4 ∑ i=1 4di ∫ T 0 ∫ Ω |∇x √ aεi |2 dxdt+ 4d5 ∫ T 0 ∫ Ω |∇x √ mε|2 dxdt + ∫ T 0 ∫ Ω (aε bε −mε/ε) (ln(aε bε)− ln(mε/ε)) dxdt 2007] QUASI–STEADY–STATE APPROXIMATION 837 First step: According to (65) (or, equivalently, to (66)) and the elementary inequality (z1 − z2) (ln z1 − ln z2) ≥ C ( √ z1 − √ z2) 2 , we get that ∫ T 0 ∫ Ω ∣ ∣ ∣ √ aεbε − √ mε/ε ∣ ∣ ∣ 2 dxdt ≤ CT , ∫ T 0 ∫ Ω ∣ ∣ ∣ √ cεdε − √ mε/ε ∣ ∣ ∣ 2 dxdt ≤ CT . Since aεi are bounded in L2, then √ aεbε (and √ cεdε ) is bounded in L2. This ensures that √ mε/ε is also bounded in L2, and hence mε/ε is bounded in L1. In particular, mε converges to 0 in L1 loc(R +;L1(Ω)). Second step: We denote by ων the set { x ∈ Ω : d(x, ∂Ω) > ν } . Let x ∈ ων (with ν positive and such that ων is not an empty set), and k ∈ R N , with | k| ≤ ν. Using now (61), we have for all i = 1, . . . , 4, ∫ T 0 ∫ ων ∣ ∣ √ aεi (t, x+ k)− √ aεi (t, x) ∣ ∣ 2 dxdt ≤ CT | k|2. Using Cauchy-Schwarz inequality and the fact that aεi is bounded in L1, we obtain ∫ T 0 ∫ ων | aεi (t, x+ k)− aεi (t, x)| dxdt = ∫ T 0 ∫ ων ∣ ∣ √ aεi (t, x+ k)− √ aεi (t, x) ∣ ∣ ∣ ∣ √ aεi (t, x+ k) + √ aεi (t, x) ∣ ∣ dxdt ≤ CT | k| . Third step: We use equations (4) and (8) in order to get ∂t ( aε + 1 2 mε ) − d1 ∆xa ε − d5 2 ∆xm ε = 1 2 (cεdε − aεbε) . Then, introducing any smooth function ϕ ≡ ϕ(x), with compact support in Ω, we have ∂t ∫ Ω ( aε + 1 2 mε ) ϕdx = d1 ∫ Ω aε∆xϕdx+ d5 2 ∫ Ω mε∆xϕdx 838 M. BISI, F. CONFORTO AND L. DESVILLETTES [December + 1 2 ∫ Ω (cεdε − aεbε) ϕdx . Therefore ∫ T 0 ∣ ∣ ∣ ∣ ∂t ∫ Ω ( aε + 1 2 mε ) ϕdx ∣ ∣ ∣ ∣ dt ≤ d1 ‖ aε‖L1 ‖∆xϕ‖L∞ + d5 2 ‖mε‖L1 ‖∆xϕ‖L∞ + 1 2 ( ‖cε‖L2 ‖dε‖L2 + ‖aε‖L2 ‖bε‖L2 ) ‖ϕ‖L∞ ≤ CT ‖ϕ‖W 2,∞ . Fourth step: We introduce ν > 0 and a mollifying sequence ϕδ(x) = δ−Nϕ(xδ ) of smooth functions with compact support B(0, δ), so that B(0, δ)+ων ⊂ Ω when δ < ν. Then, for any t ∈ [µ, T − µ] ⊂ [0, T ] (0 < µ < T/2), and for any h ∈ R, |h| ≤ µ, we get ∫ T−µ µ ∫ ων | aε(t+ h, x) − aε(t, x)| dxdt ≤ ∫ T−µ µ ∫ ων | (aε ∗x ϕδ)(t+ h, x)− (aε ∗x ϕδ)(t, x)| dxdt +2 ∫ T 0 ∫ ων | (aε ∗x ϕδ)(t, x) − aε(t, x)| dxdt ≤ ∫ T−µ µ ∫ ων ∣ ∣ ∣ ∣ [ (aε+ 1 2 mε) ∗x ϕδ ] (t+ h, x)− [ (aε+ 1 2 mε) ∗x ϕδ ] (t, x) ∣ ∣ ∣ ∣ dxdt + 1 2 ∫ T−µ µ ∫ ων | (mε ∗x ϕδ)(t+ h, x)− (mε ∗x ϕδ)(t, x)| dxdt+CT δ according to the second step. Then, thanks to the first step (i.e. to the fact that mε is of order ε in L1) and to the third step, we have ∫ T−µ µ ∫ ων | aε(t+ h, x)− aε(t, x)| dxdt ≤ ∫ T−µ t=µ ∫ x∈ων |h| ∣ ∣ ∣ ∣ ∫ 1 u=0 [∫ y∈Ω ( aε+ 1 2 mε ) (·, y)ϕδ(x− y)dy ]′ (t+ uh)du ∣ ∣ ∣ ∣ dxdt +CT ε+ CT δ 2007] QUASI–STEADY–STATE APPROXIMATION 839 ≤ |h| ∫ x∈ων ∫ 1 u=0 ∫ T t=0 ∣ ∣ ∣ ∣ ∂t ∫ y∈Ω ( aε + 1 2 mε ) (t, y)ϕδ(x− y) dy ∣ ∣ ∣ ∣ dtdudx +CT ε+CT δ ≤ CT | h| ∫ x∈ων ||ϕδ(x− ·)||W 2,∞dx+ CT ε+ CT δ. Finally, optimizing in δ, we get ∫ T−µ µ ∫ ων | aε(t+ h, x)− aε(t, x)| dxdt ≤ CT | h| δ−N−2 + CT ε+CT δ ≤ CT ( |h|1/(N+3) + ε ) . Fifth step: We use (only in this step) the notation an instead of aε in order to insist on the fact that we consider a sequence (n = 1/ε). Using the second step and the fourth step, we get (for |h| < µ, |k| < ν): ∫ T−µ µ ∫ ων | an(t+ h, x+ k)− an(t, x)| dxdt ≤ CT ( |h|1/(N+3) + |k|+ 1 n ) . (73) We introduce a mollifying sequence ψδ(t, x) = δ−(N+1) ψ(t/δ, x/δ). Let µ ∈ ]0,min {T/2, 1} [ and η > 0 be fixed. We first introduce δ > 0 such that CT (δ1/(N+3) + δ) ≤ η/3, and such that δ < µ. Then, we take m ∈ N such that CT /m ≤ η/3. Since an is a bounded sequence of L1, the sequence an ∗t,xψδ is compact in L1(]µ, T −µ[×ων). Then, it is possible (compact =⇒ uniformly bounded) to find f1, .., fP ∈ L1, such that an ∗t,x ψδ ∈ P ⋃ i=1 B(fi, η/3), where B denotes the ball B‖·‖L1(]µ,T−µ[×ων) . Thanks to (73), we see that when n ≥ m, ‖ an ∗t,x ψδ − an‖L1(]µ,T−µ[×ων) ≤ CT ( δ1/(N+3) + δ + 1 m ) ≤ 2 3 η . Then, when n ≥ m, we obtain an ∈ P ⋃ i=1 B(fi, η). Finally, for all n, an ∈ P ⋃ i=1 B(fi, η) ∪ m ⋃ i=1 B(ai, η). Conclusion: We deduce that the sequence an is uniformly bounded in 842 M. BISI, F. CONFORTO AND L. DESVILLETTES [December Remark. - We explain why we think that the assumption that the chemistry is reversible and that at most two species appear on each side of the reaction is mandatory for our analysis. Note first the crucial role played by the entropy (67), which allows to prove the bounds stated in Lemma 2.2. Since the entropy plays a key-role there, we think that our method of proof cannot be easily adapted to non reversible mechanisms of reaction. Note also that the duality argument gives a bound in L2 (lnL)2 (and not in L3), so that our method is also not adapted to reaction mechanisms involving three (or more) species on one side of a reaction. In this last situation, one can hope however that renormalized solutions (cf. [9]) could be the right concept in order to prove the validity of the QSSA. - We now discuss the initial and boundary conditions. We think that our analysis still holds if mε(0, x) = O(ε). It fails however if ε is negligible in front of mε: in such a situation, an initial layer appears and it has to be taken into account in the analysis. We also point out the fact that non bounded domains (or homogeneous Dirichlet conditions) could be considered by our method without changing much the results. However, if a non homogeneous Dirichlet condition is imposed at the boundary for mε, then some boundary layer should appear. - Finally, we discuss the type of diffusion operator that can be consid- ered. Note first that if the constant diffusion coefficients are replaced by smooth x-dependent matrices leading to a non degenerate diffusion for each equation, our analysis is still valid. We think in fact that as long as the sum of the diffusion matrices of the (non vanishing) species is non-degenerate, the validity of our analysis is preserved (cf. [9] for such a situation). Finally, adding an advection term involving a drift velocity (the velocity of the background for example, if the species that we are considering are traces) of the kind ∇x · (uai) should lead to no difficulty, provided that u is smooth enough. 3. Proof of Theorem 1.2 We now turn to the rigorous proof of the QSSA for a system of ODEs taking into account the temperature of the mixture. 2007] QUASI–STEADY–STATE APPROXIMATION 843 3.1. Proof of points 1. to 6. of Theorem 1.2 We begin with the proof that (48)-(50) holds on [0, Tε[, where Tε is the largest possible time of existence of a solution of the Cauchy problem (33)-(38) whose components are all strictly positive. More precisely, we define first τε as the largest possible time of existence of a solution of the Cauchy problem (33)-(38). This quantity is well-defined and strictly positive thanks to Cauchy-Lipschitz’ theorem, which can be used since all the functions in the system are assumed to be of class C1 at least in a neighborhood of the initial datum. Then, we define Tε=sup{T ∈ [0, τε[ : ∀t∈]0, T ], nεA,B,C,D,M(t)>0,Θε(t)> 0}. It is obvious that Tε > 0 if all initial data are strictly positive. Then, if nC0 = 0, one can verify that (nεC) ′(0) = 0 and (nεC) ′′(0) > 0. The same is true when C is replaced by D. As a consequence, Tε > 0 also in this case. At this point, it can be easily proven that no one of the field variables can vanish at a time T ∗ < τε. In fact, if we consider the first time when at least one of the field variables vanishes, all possible cases lead to one of the following two contradictions : 1. In some cases, we get that one of the vanishing quantities has strictly positive prime derivative at time T ∗, and this is in contradiction with the fact that it is strictly positive before T ∗, 2. In other cases, we get that the unique solution is the constant one, and this leads to the contradiction that some of the variables vanishing at T ∗ are supposed to have strictly positive initial values. Therefore, we must have Tε = τε. Remember that the chemical bond energies EA, . . . , ED are assumed to be all positive. The independent conserved quantities on [0, Tε] are: nεA + nεC + nεM := n̄1 > 0 , (78) nεA + nεD + nεM := n̄2 > 0 , (79) nεB + nεC + nεM := n̄3 > 0 , (80) EAn ε A + EBn ε B + ECn ε C + EDn ε D + ( EA + EB + 1 ε ) nεM + 3 2 nεΘε := W̄ > 0 . (81) 844 M. BISI, F. CONFORTO AND L. DESVILLETTES [December Therefore, it is possible to obtain (up to time Tε) estimates (48), (49) and the upper bound in (50) : Θε(t) ≤ c5. Finally, in order to get a lower bound for Θε, we introduce the entropy eε = nεA lnnεA + nεB lnnεB + nεC lnnεC + nεD lnnεD + nεM lnnεM − 3 2 nε lnΘε, which is easily seen to be a decreasing function of time. In fact, one can check that (eε)′ = −ϕε Sε1 { ln (nεAn ε B)− ln ( nεMe 1 εΘε+ 3 2 lnΘε− 5 2 )} −ψε Sε2 { ln (nεCn ε D)− ln ( nεMe 1 εΘε− ∆E Θε + 3 2 lnΘε− 5 2 )} ≤ 0, thanks to the standard inequality (x− y) (lnx− ln y) ≥ 0, ∀x, y > 0. So, using the lower bound for nε and the inequality x lnx > −1, ∀x > 0, the lower bound for Θε in (50) (on [0, Tε[) follows from the estimate eε (t) ≤ e (0) ∀ t ∈ [0, T ε[. According to standard theorems for ODEs, we deduce from the previous properties of boundedness that Tε(= τε) = +∞. Notice also that the properties of existence and uniqueness of solutions (together with boundedness and strict positivity of the components of the solution) for the limiting system can be proven following the same lines as above. The fact that the constants in (52)-(54) are the same as those in (48)-(50) is a consequence of point 7. proven below. 3.2. Estimate for n ε M At this point, we are able to prove the statement (55) of Theorem 1.2. Note first that eq. (41) can be rewritten under the form ( nεMe ∫ · 0 ζε(s) ds )′ = e ∫ · 0 ζε(s) ds (ϕεnεAn ε B + ψεnεCn ε D) , 2007] QUASI–STEADY–STATE APPROXIMATION 847 3.3. Conclusion of the proof of Theorem 1.2 In order to prove (56), (57), we subtract from each equation (39), (40), (42) for the functions nεA, n ε B, n ε C , n ε D and Θε the corresponding ones (43)- (45) for nA, nB, nC , nD and Θ. We obtain, in matrix form, that    nεA,B − nA,B nεC,D − nC,D 3 2n εΘε − 3 2nΘ    ′ =   1 −1 ∆E   (F ε − F ) +     − ϕε ϕε+ψεe−∆E/Θε − ψεe−∆E/Θε ϕε+ψεe−∆E/Θε (EA+EB)ϕε+(EC+ED)ψεe−∆E/Θε ϕε+ψεe−∆E/Θε     (nεM )′ −   0 0 EA + EB + 1 ε   (nεM )′. Integrating this identity on [0, t], recalling that the initial values do not depend on ε, and that nεM (0) = nM (0) = 0, we get :    nεA,B − nA,B nεC,D − nC,D 3 2n εΘε − 3 2nΘ    (t) = ∫ t 0   1 −1 ∆E   (F ε−F ) (s)ds+     − ϕε ϕε+ψεe−∆E/Θε − ψεe−∆E/Θε ϕε+ψεe−∆E/Θε (EA+EB)ϕε+(EC+ED)ψεe−∆E/Θε ϕε+ψεe−∆E/Θε     (t)nεM (t) − ∫ t 0     − ϕε ϕε+ψεe−∆E/Θε − ψεe−∆E/Θε ϕε+ψεe−∆E/Θε (EA+EB)ϕε+(EC+ED)ψεe−∆E/Θε ϕε+ψεe−∆E/Θε     ′ (s)nεM (s)ds −   0 0 EA + EB + 1 ε   nεM (t) . The last term of the right-hand side of this equation is clearly bounded by C e − 1 c4ε ε . 848 M. BISI, F. CONFORTO AND L. DESVILLETTES [December Noticing that ϕε + ψε e− ∆E Θε ≥ infc4≤Θ≤c5 ϕ(Θ), the second term of the right-hand side is bounded by C e − 1 c4ε . The third term can be written under the form ∫ t 0 ((Θ ε)′R(Θε)nεM )(s) ds, where (thanks to the assumption of smoothness of ϕ and ψ) R(Θ) ≤ C for Θ ∈ [c4, c5]. Then, we notice that Θε satisfies the equation (Θε)′ = ( Θε − 2 3ε ) ϕε nε Sε1 + ( Θε + 2 3 (∆E − 1 ε ) ) ψε nε Sε2, so that thanks to (85) and (49), (50), |(Θε)′(t)| ≤ C ε . Finally, we see that the third term of the right-hand side is bounded by C e − 1 c4ε ε t. Now, thanks to the hypothesis of smoothness of ϕ and ψ, and hence of F , and thanks to estimates (49), etc., we can write that ( |nεA − nA|+ · · ·+ |nεD − nD|+ ∣ ∣ ∣ ∣ 3 2 nεΘε − 3 2 nΘ ∣ ∣ ∣ ∣ ) (t) ≤ sup(1,∆E) ∫ t 0 ( ∫ 1 0 ∣ ∣ ∣ ∣ ∇F ( (1−u)nεA+unA, . . . , (1−u) 3 2 nεΘε+u 3 2 nΘ )∣ ∣ ∣ ∣ du × ( |nεA − nA|+ · · · + ∣ ∣ ∣ ∣ 3 2 nεΘε − 3 2 nΘ ∣ ∣ ∣ ∣ )) (s) ds + C ε e − 1 c4 ε (1 + t) ≤ C ∫ t 0 ( |nεA − nA|+ · · ·+ ∣ ∣ ∣ ∣ 3 2 nεΘε − 3 2 nΘ ∣ ∣ ∣ ∣ ) (s) ds+ C ε e − 1 c4 ε (1 + t). Finally, using Gronwall’s lemma, we deduce that (for t ∈ [0, T ]) ( |nεA − nA|+ · · ·+ |nεD − nD|+ ∣ ∣ ∣ ∣ 3 2 nεΘε − 3 2 nΘ ∣ ∣ ∣ ∣ ) (t) ≤ C 1 + T ε e − 1 c4 ε eC T . Theorem 1.2 is then easily deduced by noticing that nε and n are bounded below and above. Remark. - Note first that the restrictions on the sign of ∆E, or on the strict posi- tivity of nA0 and nB0, can easily be removed (up to very small changes in the statement of the theorem). Then, any reasonable energy law (for example the energy law of perfect polytropic gases) could be handled by 2007] QUASI–STEADY–STATE APPROXIMATION 849 our approach: the main modification is that one should write the correct entropy associated with the energy law under consideration. Finally, as in the case of reaction-diffusion, we think that our method (in which the entropy plays a crucial role) is not adapted to treat irreversible problems. - As in the previous section, our analysis still holds if mε(0, x) = O(ε), but it fails if ε is negligible in front of mε: in such a situation, an initial layer appears. - Our estimates of convergence are uniform on any compact set (in time) of R. If one wishes to obtain uniformity with respect to time on R, one should study the large time behavior of the systems when ε > 0 and ε = 0, and use the fact that the mass action law defining the equilibrium when ε = 0 is a consequence of the mass action law for a given ε > 0, while the other constraints (conservation of the number of molecules and energy) at ε = 0 are the limit of the corresponding expression when ε > 0. Acknowledgment The help of the Franco-italian GdR GREFI-MEFI for the preparation of this work is acknowledged. References 1. A. Blouza, F. Coquel and F. Hamel, Reduction of linear kinetic systems with multiple scales, Combust. Theory Modelling, 4(2000), 339-362. 2. D. Bothe, The instantaneous limit of a reaction–diffusion system, in: G. Lumer, L. Weis Eds., Evolution Equations and Their Applications in Physical and Life Sciences, in: Pure and Applied Mathematics, Vol. 125, Dekker, (2001), 215-224. 3. D. Bothe, Instantaneous limits of reversible chemical reactions in presence of macroscopic convection, J. Differential Equations, 193(2003), 27-48. 4. D. Bothe and D. Hilhorst, A reaction–diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286(2003), 125-135. 5. H. Brezis, Analyse Fonctionnelle: Theorie et Applications, Masson, Paris, 1983. 6. G.M. Come, Radical reaction mechanisms. Mathematical theory, J. Phys. Chem., 81 (1977), 2560-2563. 7. L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319(2006), no. 1, 157-176.