EXPLICIT EXPONENTIAL CONVERGENCE TO ... - Uni Graz

EXPLICIT EXPONENTIAL CONVERGENCE TO EQUILIBRIUM FOR NONLINEAR REACTION-DIFFUSION SYSTEMS WITH DETAILED BALANCE CONDITION KLEMENS FELLNER, BAO QUOC TANG

Abstract. The convergence to equilibrium of mass action reaction-diffusion systems arising from networks of chemical reactions is studied. The considered reaction networks are assumed to satisfy the detailed balance condition and have no boundary equilibria. We propose a general approach based on the so-called entropy method, which is able to quantify with explicitly computable rates the decay of an entropy functional in terms of an entropy entropy-dissipation inequality based on the totality of the conservation laws of the system. As a consequence follows convergence to the unique detailed balance equilibrium with explicitly computable convergence rates. The general approach is further detailed for two important example systems: a single reversible reaction involving an arbitrary number of chemical substances and a chain of two reversible reactions arising from enzyme reactions.

Contents 1. Introduction and main results 2. Mathematical settings and the general method 2.1. Mathematical settings 2.2. A constructive method to prove the EED estimate 3. A single reversible reaction - Proof of Theorem 1.1 4. Reversible enzyme reactions - Proof of Theorem 1.2 5. Summary, further applications and open problems 5.1. Further applications 5.2. Open problems References

1 7 7 11 17 22 27 27 28 28

1. Introduction and main results In this paper, we study the quantitative convergence to equilibrium for a class of reaction-diffusion systems arising from chemical reaction networks. The considered reaction-diffusion systems describe networks of chemical reactions according to the law of mass action kinetics and under the assumption of a detailed balance condition. More precisely, we consider I chemical substances A1 , . . . , AI reacting in R reversible reactions of the form α1r A1 + . . . + αIr AI

kr,b kr,f

β1r A1 + . . . + βIr AI

for r = 1, 2, . . . , R with the nonnegative stoichiometric coefficients αr = (α1r , . . . , αIr ) ∈ ({0} ∪ [1, ∞))I and β r = (β1r , . . . , βIr ) ∈ ({0} ∪ [1, ∞))I and the positive forward and backward reaction rate constants kr,f > 0 and kr,b > 0. The corresponding reaction-diffusion system for the concentration vector c = (c1 , . . . , cI ) : Ω × R+ → [0, +∞)I reads as ∂ c = div(D∇c) − R(c), ∂t ∇c · ν = 0, c(x, 0) = c0 (x),

in Ω × R+ , on ∂Ω × R+ ,

(1.1)

on Ω,

2010 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92. Key words and phrases. Reaction-Diffusion Systems; Exponential Convergence to Equilibrium; Entropy Method; Chemical Reaction Networks; Detailed Balance Condition. 1

2

K. FELLNER, B.Q. TANG

where Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω and normalized volume, i.e. |Ω| = 1, D = diag(d1 (x), . . . , dI (x)) is the uniformly positive definite diffusion matrix, i.e. 0 < di,min ≤ di (x) ≤ di,max for all i = 1, . . . , I and all x ∈ Ω, and the reaction vector R(c) represents the chemical reactions according to the mass action law, i.e. R(c) =

R X

kr,f c

αr

− kr,b c

βr

r

r

(α − β )

with

r=1

c

αr

=

I Y

αr

ci i

for r = 1, 2, . . . , R.

i=1

Denote by m = codim(span{αr − β r : r = 1, 2, . . . , R}). If m > 0, there exists a matrix Q ∈ Rm×I of zero left-eigenvectors such that Q R(c) = 0 for all states c. Thus, we have the following conservation laws for (1.1) Z Z Q c(t)dx = Q c0 dx or equivalently Q c(t) = M := Q c0 for all t > 0, Ω

Ω

R where c = (c1 , . . . , cI ) with ci = Ω ci (x)dx is the spatially averaged concentration vector (recall |Ω| = 1) and M ∈ Rm + denotes the vector of positive initial masses. The large time behaviour of solutions to nonlinear reaction-diffusion systems is a highly active research area, which poses many open problems. Classical analytical methods include e.g. linearisation techniques, spectral analysis, invariant regions and Lyapunov stability arguments. More recently, the so-called entropy method proved to be a very useful and powerful improvement of classical Lyapunov methods, as it allows, for instance, to show explicit exponential convergence to equilibrium for reaction-diffusion systems. The basic idea of the entropy method consists in studying the large-time asymptotics of a dissipative PDE model by looking for a nonnegative convex entropy functional E(f ) and its nonnegative entropy-dissipation functional d E(f (t)) dt along the flow of a PDE model, which is well-behaved in the following sense: firstly, all states satisfying D(f ) = 0 as well as all the involved conservation laws identify a unique entropy-minimising equilibrium f∞ , i.e. D(f ) = 0 and conservation laws ⇐⇒ f = f∞ , D(f ) = −

and secondly, there exists an entropy entropy-dissipation (EED for short) estimate of the form D(f ) ≥ Φ(E(f ) − E(f∞ )),

Φ(x) ≥ 0,

Φ(x) = 0 ⇐⇒ x = 0,

for some nonnegative function Φ. We remark, that such an inequality can only hold when all the conserved quantities are taken into account. Moreover, if Φ0 (0) 6= 0, a Gronwall argument usually implies exponential convergence toward f∞ in relative entropy E(f ) − E(f∞ ) with a rate, which can be explicitly estimated. Finally, by applying Csiszár-Kullback-Pinsker type inequalities to the relative entropy E(f ) − E(f∞ ) (recall that E(f ) is convex), one obtains exponential convergence to equilibrium, for instance, w.r.t. the L1 -norm. The entropy method is a fully nonlinear alternative to arguments based on linearisation around the equilibrium and has the advantage of being quite robust with respect to variations and generalisations of the model system. This is due to the fact that the entropy method relies mainly on functional inequalities which have no direct link to the original PDE model. Generalised models typically feature related entropy and entropy-dissipation functionals and previously established EED estimates may very usefully be re-applied. The entropy method has previously been used for scalar equations: nonlinear diffusion equations (such as fast diffusions [CV03, PD02], Landau equation [DV00]), integral equations (such as the spatially homogeneous Boltzmann equation [TV99, TV00, Vil03]), kinetic equations (see e.g. [DV01, DV05, FNS04]), or coagulation-fragmentation models (see e.g. [CDF08, CDF08a]). For certain systems of driftdiffusion-reaction equations in semiconductor physics, an entropy entropy-dissipation estimate has been shown in two dimensions indirectly via a compactness-based contradiction argument in [GGH96, GH97]. Some first results for EED estimates for reaction-diffusion systems with explicit rates and constants were established in [DF06, DF08] in the case of particular reversible equations with quadratic nonlinearities. In this paper, we shall generalise the entropy method to reaction-diffusion systems with arbitrary mass action law nonlinearities and, as a consequence, show explicit exponential convergence to equilibrium for (1.1). Thus, the results in this paper significantly improve the recent works on convergence to equilibrium

CONVERGENCE TO EQUILIBRIUM FOR REACTION-DIFFUSION SYSTEMS

3

for chemical reaction-diffusion systems, see e.g. [DF06, DF08, DFM08, GZ10, DF14, MHM15, FL16, FLT]. The analysis in this work uses the detailed balance condition, which also allows to assume (without loss of generality due to a suitable scaling argument) that kr,f = kr,b = kr > 0

for all r = 1, 2, . . . , R.

The key quantity of our study is the logarithmic entropy (free energy) functional I Z X E(c) = (ci log ci − ci + 1) dx, i=1

Ω

which decays monotone in time, thanks to the detailed balance condition, according to the following entropy-dissipation functional Z I Z R X X r r r r |∇ci |2 d dx + di (x) kr (cα − cβ )(log cα − log cβ )dx ≥ 0. D(c) = − E(c) = dt ci Ω r=1 i=1 Ω For a fixed positive initial mass vector M ∈ Rm + , we denote by c∞ the detailed balance equilibrium of (1.1) with mass M, that is the unique vector of positive constants c∞ > 0, which balances all the reactions, i.e. r βr cα for all r = 1, 2, . . . , R ∞ = c∞ , and satisfies the mass conservation laws Q c∞ = M. The key step of the entropy method in order to prove exponential convergence to equilibrium of (1.1) is the following EED estimate (1.2) D(c) ≥ λ(E(c) − E(c∞ )) for all c ∈ L1 (Ω; [0, +∞)I ) obeying the mass conservation Q c = M, where λ = λ(Ω, D, M, αr , β r ) is a constant depending on the domain Ω, the diffusion coefficients D, the initial mass vector M and the stoichiometric coefficients. Once such a functional inequality is proved, applying it to (suitable strong or weak) solutions of the reaction-diffusion system yields via a Gronwall argument exponential convergence in relative entropy with the rate λ, which can be explicitly calculated. Moreover, by applying a Csiszár-Kullback-Pinsker type inequality one obtains L1 -convergence to equilibrium of solutions to (1.1) with the rate e−λt/2 . In [MHM15], by using an inspired convexification argument, the authors proved that such a λ > 0 always exists for system (1.1) under the detailed balance condition. Moreover, they gave an explicit bound of λ in the case of the quadratic reaction 2X Y . However, because of the non-convex structure of the problem, obtaining estimates on λ seems difficult in the case of more than two substances, e.g. for systems like αA1 + βA2 γA3 or A1 + A2 A3 + A4 . Inspired by various ideas from [DF08, DF14, FL16, FLT], this work aims to propose a constructive way to prove quantitatively the EED estimate (1.2) for general mass action law reaction-diffusion systems. The main advantage of our method is that, by extensively using the structure of the mass conservation laws, the proof relies on elementary inequalities and has the advantage of providing explicit estimates for the convergence rate λ. Another novelty is that the here-proposed method is robust in the sense that it is also applicable to chemical reaction networks where substances are supported on different compartments (see subsection 5.1 for a specific example of a volume-surface reaction-diffusion system). We remark that the method of convexification as presented in [MHM15] seems not to apply to such reaction networks. In the following we shall sketch the main ideas of our method to prove (1.2) by dividing the proof into four steps. These steps are designed as a chain of estimates, which at the end allows to take into account the conservation laws, which are crucial to the validity of (1.2): Step 1: We use an additivity property of the logarithmic entropy in order to split the right hand side of (1.2) into two parts E(c) − E(c∞ ) = E(c) − E(c) + E(c) − E(c∞ ) , where the first part E(c)−E(c) can be controlled by (a part of) D(c) via the Logarithmic Sobolev Inequality and the second part E(c) − E(c∞ ) contains only spatially averaged terms, which are however still subject to the full dynamics of reactions and diffusion.

4


Step 2: We estimate (the remaining part of) D(c) as well as E(c) − E(c∞ ) in terms of quadratic forms of the square roots of the concentrations, which are significantly easier to deal with. By using √ capital letters as short hand notation for the square roots of various quantities, i.e. Ci = ci √ and Ci,∞ = ci,∞ and some elementary inequalities, we have (see Section 2.2 for details) I R

r X X r 2 1

D(c) ≥ 2 di,min k∇Ci k2L2 (Ω) + 2 kr Cα − Cβ 2 , 2 L (Ω) r=1 i=1

and 2 I q X 2 E(c) − E(c∞ ) ≤ K2 Ci − Ci,∞ . i=1

Step 3: In order to be able to use the constrains provided by the conservation laws, we bound the reaction term of D(c) below by a reaction term of the corresponding spatial averages, i.e. I R

r X X r 2 1

D(c) ≥ 2di,min k∇Ci k2L2 (Ω) + 2 kr Cα − Cβ 2 2 L (Ω) r=1 i=1 ! I R X X αr βr 2 C −C ≥ K3 k∇Ci k2L2 (Ω) + r=1

i=1

where C = (C1 , . . . , CI ) with Ci =

R Ω

Ci (x)dx.

Step 4: As a final step, we are left to find a constant K1 > 0 such that ! 2 I q I R X X K2 X αr βr 2 2 C −C ≥ K1 Ci2 − Ci,∞ . k∇Ci kL2 (Ω) + K3 i=1 r=1 i=1

(1.3)

To prove this claim, we will employ a change of variable, which allows to quantify the conservation laws in terms of deviations around the equilibrium values, i.e. 2 Ci2 = Ci,∞ (1 + µi )2 ,

µi ∈ [−1, +∞).

(1.4)

A key observation is that the non-negativity of the concentration vector c provides a natural lower bound µi ≥ −1, while the conservation laws Q C2 = Q C2∞ impose also certain upper bounds on the new variable µi . Then, the proof of (1.3) distinguishes two cases: i) when all Ci2 are strictly bounded away from zero and ii) the case of ”degenerate states” when at least one Ci20 is sufficiently ”small”, which means that there is only a sufficiently small amount of mass of the species Ai0 present. In the former case, the ansatz (1.4) allows to reduce (1.3) into a finite dimensional inequality in terms of the new variables µ1 , . . . , µI under the constraints of the conservation laws. In the latter case, if some Ci20 is much smaller than e.g. its equilibrium value, we are able to quantitatively estimate that such ”degenerate states” are far away from equilibrium in the sense that the left hand side of (1.3) is always bounded below by a positive constant, which is again expressed in terms of the conservation laws. Thus, one obtains (1.3) by choosing a suitable K1 after observing the fact that the right hand side of (1.3) is naturally bounded above by a constant. We remark that the Steps 1, 2 and 3 can be proved without using the conservation laws. Hence, we are able to prove these three steps in full generality. Step 4, however depends on the structure of conservation laws defined by the matrix Q. And because the matrix Q is in general not explicitly given, this prevents an entirely explicit proof of this step in the general case. Nevertheless, we shall illustrate for two important specific systems how our method allows to make Step 4 entirely explicit once the matrix Q is explicitly given. Before stating our results for those examples, let us remark that the question of global existence of (classical, strong or weak) solutions is far open for general nonlinear reaction-diffusion systems, in particular for systems of type (1.1) (see e.g. the survey [Pie10]). This is due to the lack of sufficiently strong a-priori estimates (comparison principles do not hold except for special systems) in order to control the nonlinear terms. Recently, Fischer [Fis15] proved the global existence of so-called ”renormalised solution” for (1.1). All the estimates presented in our paper hold for renormalised solution. Indeed, it is shown in [Fis15] 1 that ci log ci ∈ L∞ loc ([0, +∞); L (Ω)) for all i = 1, 2, . . . , I, which makes the entropy functional E(c) well defined.


5

In this paper, we will detail the proposed strategy for two important specific models: a general single reversible reaction with arbitrary number of substances α1 A1 + . . . + αI AI β1 B1 + . . . + βJ BJ

(1.5)

and a chain of two reversible reactions, which generalises the Michaels-Menton model for catalytic enzyme kinetics (see e.g. [Mur02]) A1 + A2 A3 A4 + A5 . (1.6) Note that with for the single reversible reaction (1.5), it is more convenient and usual in the literature to change the notation compared to the general system (1.1) by splitting the concentration vector c into a left-hand-side and a right-hand-side-concentration vector, i.e. c = (c1 , . . . , cI ) → (a, b) = (a1 , . . . , aI , b1 , . . . , aJ ), where I denotes now the number of left-hand-side concentrations and J the number of right-hand-side concentrations. This notation allows a clearer presentation of the corresponding system and the proofs. At first, after assuming that the forward and backward reaction rate constants are normalised to one, the mass action reaction-diffusion system modelling (1.5) reads as  β α in Ω × R+ , i = 1, 2, . . . , I,  ∂t ai − div(da,i (x)∇ai ) = −αi (a − b ),  ∂ b − div(d (x)∇b ) = −β (aα − bβ ), in Ω × R+ , j = 1, 2, . . . , J, t j b,j j j (1.7) ∇ai · ν = ∇bj · ν = 0, on ∂Ω × R+ , i = 1, . . . , I, j = 1, . . . , J,    a(x, 0) = a0 (x), b(x, 0) = b0 (x), in Ω, where a = (a1 , . . . , aI ) and b = (b1 , . . . , bJ ) denote the two vectors for left- and right-hand side concentrations and α = (α1 , . . . , αI ) ∈ [1, ∞)I and β = (β1 , . . . , βJ ) ∈ [1, ∞)I are the positive vectors of QI i the stoichiometric coefficients assossiated to the single reaction (1.5). Recall that aα = i=1 aα i and Q J β bβ = i=1 bi i . This system (1.7) possesses the following IJ mass conservation laws ai bj + = Mi,j , αi βj

i = 1, . . . , I, j = 1, . . . , J,

(1.8)

from which exactly m = I + J − 1 conservation laws are linear independent. That means the matrix Q in this case has the dimension Q ∈ R(I+J−1)×(I+J) . See Lemma 3.1 for an explicit form of Q. After choosing and fixing I + J − 1 linear independent components from the IJ conserved initial masses (Mi,j ) ∈ RIJ + , we denote by M = (Mi,j ) ∈ RI+J−1 the vector of initial masses corresponding to the selected I + J − 1 I+J of (1.7) is coordinates of (Mi,j ) ∈ RIJ + . The unique detailed balance equilibrium (a∞ , b∞ ) ∈ R+ defined by   ai,∞ + bj,∞ = Mi,j ∀i = 1, 2, . . . , I, ∀j = 1, 2, . . . , J, αi βj aα = bβ . ∞ ∞ The corresponding entropy functional and entropy-dissipation for system (1.7) are J Z I Z X X (bj log bj − bj + 1)dx E(a, b) = (ai log ai − ai + 1)dx + i=1

Ω

j=1

(1.9)

Ω

and D(a, b) =

I Z X i=1

Ω

da,i (x)

Z J Z X |∇ai |2 |∇bj |2 aα dx + db,j (x) dx + (aα − bβ ) log β dx, ai bj b Ω j=1 Ω

(1.10)

respectively. I+J−1 Theorem 1.1 (Explicit convergence to equilibrium). Let M ∈ R+ be a fixed positive initial mass vector corresponding to I + J − 1 linear independent conservation laws (1.8). Denote by (a∞ , b∞ ) the detailed balance equilibrium of (1.7). Then, for any nonnegative (a, b) ∈ L1 Ω; [0, ∞)I+J satisfying the mass conservation laws (1.8), we have D(a, b) ≥ λ(E(a, b) − E(a∞ , b∞ )) (1.11) with E(a, b) and D(a, b) defined in (1.9) and (1.10) respectively, where the constant λ > 0 can be explicitly estimated in terms of the initial mass M, the domain Ω, the positive stoichiometric coefficients α, β and the diffusion coefficients da,i and db,j .

6


Consequently, the solution to (1.7) obeys the following exponential convergence to equilibrium I X

ai,∞ k2L1 (Ω)

kai (t) −

i=1

+

J X

−1 kbj (t) − bj,∞ k2L1 (Ω) ≤ CCKP (E(a0 , b0 ) − E(a∞ , b∞ ))e−λt

j=1

where CCKP is the constant in a Csisz´ ar-Kullback-Pinsker inequality in Lemma 2.4. Remark 1.1. Theorem 1.1 generalises the previous results of [DF06, DF08, GZ10, MHM15, FL16], where only special cases of system (1.7) were treated. As second example, after assuming that all the forward and backward reaction rate constants are normalised to one, the reaction-diffusion system modelling (1.6) reads (again in the general notation of system (1.1)) as  ∂t c1 − div(d1 (x)∇c1 ) = −c1 c2 + c3 , in Ω × R+ ,      ∂ c − div(d (x)∇c ) = −c c + c , in Ω × R+ , t 2 2 2 1 2 3     ∂t c3 − div(d3 (x)∇c3 ) = c1 c2 + c4 c5 − 2c3 , in Ω × R+ ,  (1.12) ∂t c4 − div(d4 (x)∇c4 ) = −c4 c5 + c3 , in Ω × R+ ,    ∂t c5 − div(d5 (x)∇c5 ) = −c4 c5 + c3 , in Ω × R+ ,      ∇ci · ν = 0, i = 1, 2, . . . , 5, on ∂Ω × R+ ,    ci (x, 0) = ci,0 (x), in Ω. The mass conservation laws of (1.12) are ci + c3 + cj = Mi,j ,

∀i ∈ {1, 2}

and

∀j ∈ {4, 5}

(1.13) 3×5

and among these there are precisely m = 3 linear independent conservation laws, thus Q ∈ R . In the following, we denote by c = (c1 , . . . , c5 ) the concentration vector and by (Mi,j ) = (M1,4 , M1,5 , M2,4 , M2,5 ) ∈ R4 the initial mass vector. Note that the initial mass vector M is determined by any three coordinates corresponding to three linear independent conservation laws and by a fixed initial mass vector (Mi,j ) ∈ R4+ we mean that these three coordinates are given and the remaining coordinates are subsequently calculated. The unique detailed balance equilibrium c∞ = (c1,∞ , . . . , c5,∞ ) ∈ R5 to (1.12) is defined by   ∀i ∈ {1, 2} and ∀j ∈ {4, 5}, ci,∞ + c3,∞ + cj,∞ = Mi,j , c1,∞ c2,∞ = c3,∞ ,   c4,∞ c5,∞ = c3,∞ . The corresponding entropy functional and entropy-dissipation for system (1.12) are 5 Z X E(c) = (ci log ci − ci + 1)dx i=1

(1.14)

Ω

and D(c) =

5 Z X i=1

|∇ci |2 di (x) dx + ci Ω

Z c1 c2 c4 c5 (c1 c2 − c3 ) log + (c4 c5 − c3 ) log dx, a3 c3 Ω

(1.15)

respectively. Theorem 1.2 (Explicit convergence to equilibrium). Let M ∈ R4+ be a fixed positive initial mass vector corresponding to three linear independent conservation laws of (1.12). Denote by c∞ the corresponding unique detailed balance equilibrium of (1.12). Then, for any nonnegative measurable function c = (c1 , . . . , c5 ) ∈ L1 Ω; [0, +∞)5 satisfying the mass conservation laws (1.13), we have D(c) ≥ λ(E(c) − E(c∞ )) with E(c) and E(c) are defined in (1.14) and (1.15) respectively, where λ > 0 is a positive constant which can be explicitly estimated in terms of the initial mass M, the domain Ω and the diffusion coefficients di , i = 1, 2, . . . , 5. As a consequence, the solutions c = (c1 , . . . , c5 ) to (1.12) converge exponentially to the equilibrium defined by its initial mass, 5 X

−1 kci (t) − ci,∞ k2L1 (Ω) ≤ CCKP (E(c0 ) − E(c∞ ))e−λt ,

i=1

where CCKP is the constant in the Csisz´ ar-Kullback-Pinsker inequality.

∀t > 0,


7

The rest of this paper is organized as follows: In Section 2, we give the details of the general mathematical settings and the constructive method containing the mentioned four steps. Also in this section, the Steps 1, 2 and 3 will be proved rigorously and explicitly in the general case. The proofs of Theorems 1.1 and 1.2 are presented in Sections 3 and 4 respectively. Finally, we discuss the further possible applications and some open problems in Section 5. 2. Mathematical settings and the general method In this section, we first briefly recall the mathematical settings of mass action reaction-diffusion systems modelling chemical reaction networks and then give the details of the proposed method. 2.1. Mathematical settings. For convenience to the reader, we will adopt the notations from [MHM15] and refer also to e.g. [VVV94]. Consider I species A1 , . . . , AI reacting via R reversible reactions according to the mass-action law of the form: kr,f

α1r A1 + . . . + αIr AI

(2.1)

β1r A1 + . . . + βIr AI

kr,b

for r = 1, 2, . . . , R, where R ∈ N, αr = (α1r , . . . , αIr ) ∈ ({0} ∪ [1, +∞))I and β r = (β1r , . . . , βIr ) ∈ ({0} ∪ [1, +∞))I are the vectors of nonnegative stoichiometric coefficients and kr,b , kr,f are the backward and forward reaction rate constants. Denote by c(t, x) ∈ RI the vector of concentrations, then the reaction-diffusion process is modeled by the semilinear parabolic PDE system ∂ c = div(D∇c) − R(c) in Ω × R+ and ∇c · ν = 0 in ∂Ω × R+ , (2.2) ∂t subject to nonnegative initial data c(x, 0) = c0 (x), x ∈ Ω, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω which has the outward normal unit vector ν. Note that without loss of generality, we can rescale the spatial variable such that the volume of Ω is normalised, i.e. |Ω| = 1. The diffusion matrix D(x) = diag(di (x))i=1,...,I is diagonal and positive definite. We assume moreover that the diffusion coefficients satisfy 0 < di,min ≤ di (x) ≤ di,max

∀x ∈ Ω,

∀i = 1, 2, . . . , I.

(2.3)

The reaction vector R models the reactions (2.1) according to the mass action law R(c) =

R X

r

kr,f cα − kr,b cβ

r

(αr − β r )

with

r

cα =

r=1

I Y

αr

ci i .

i=1

∈ ({0} ∪ [1, +∞))I and To determine the mass conservation laws for (2.2), we write α = r I r r β = (β1 , . . . , βI ) ∈ ({0} ∪ [1, +∞)) as columns, which gives the stoichiometric matrix r

(α1r , . . . , αIr )

>

W = ((β r − αr )r=1,...,R ) ∈ RR×I , which is also called Wegscheider matrix. Note that according to the mass action law, we can write R(c) in the form R(c) = −W > K(c), (2.4) where r r K(c) = (Kr (c))r=1,...,R := kr,f cα − kr,b cβ

r=1,...,R

denotes the vector of reaction rates according to the mass action law. The range rg(W > ) is called the stoichiometric subspace and due to (2.4) we have R(c) ∈ rg(W > ). We can now determine the mass conservation laws as follows: denote by m = dim ker(W ), the codim of W . • If m > 0 we choose a matrix Q ∈ Rm×I such that rankQ = m and Q W > = 0, i.e. the rows of Q form a basis of ker(W ). Since R(c) ∈ rg(W > ), we have Q R(c) = 0

for all c ∈ RI .

(2.5)

By denoting (with |Ω| = 1) Z c = (c1 , . . . , cI )

with ci =

ci (x)dx Ω

(2.6)

8


and using the no-flux boundary condition for c on ∂Ω, we end up with the conservation laws Z Z Z d Q c(t) dx = QD ∇c · ν dS − QR(c) dx = 0 (2.7) dt Ω ∂Ω Ω or equivalently Q c(t) = Q c(0) =: M ∈ Rm

for all

t > 0,

(2.8)

where M denotes the initial mass vector. • If m = 0 then the system (2.2) has no conservation law. For physical reasons, we only consider nonnegative concentrations as solutions. It is well-known (see e.g. [Pie10]) that reaction vector R(c) satisfies the following quasi-positivity condition: If R(c) = (R1 (c), . . . , RI (c))> then Ri (c1 , . . . , ci−1 , 0, ci+1 , . . . , cI ) ≥ 0

∀i = 1, 2, . . . I with c1 , . . . , ci−1 , ci+1 , . . . , cI ≥ 0,

which is naturally satisfied by mass action law reactions of the form Ri (c) =

R X

r

kr,f cα − kr,b cβ

r

(αir − βir )

r=1

for i = 1, 2, . . . , I. Thus, we have (see e.g. [Pie10]) Lemma 2.1 (Nonnegativity). If the initial concentration vector c0 is nonnegative, then the solution vector c(t) remains nonnegative for all t > 0. ∗ I Definition 2.1 (Equilibrium). Fix a positive initial mass vector M ∈ Rm + . A state c ∈ [0, +∞) is called a homogeneous equilibrium (or shortly an equilibrium) for (2.2) if and only if

R(c∗ ) = 0

and

Q c∗ = M.

To study the large time behaviour of (2.2), we impose the following crucial assumptions: (A1) System (2.2) satisfies a detailed balance condition, that is, there exists an equilibrium c∞ ∈ (0, +∞)I such that r βr kr,f cα ∀r = 1, 2, . . . , R. ∞ = kr,b c∞ , This equilibrium c∞ is called a detailed balance equilibrium. (A2) There is no boundary equilibrium, that is (2.2) does not possess an equilibrium lying on ∂[0, +∞)I . Therefore any equilibrium c∞ = (c1,∞ , . . . , cI,∞ )> to (2.2) satisfies ci,∞ > 0 for all i = 1, 2, . . . , I. Remark 2.1. • The assumption (A1) allows to rescale the system such that we can assume kr,f = kr,b = kr for all r = 1, 2, . . . , R. Thus, the reaction rate constant of each reaction is equal to the reaction rate constant of the reverse reaction. This helps us to see that the logarithmic (free energy) entropy functional (see (2.1)) is a Lyapunov functional, i.e. it decreases in time along the trajectories of the system (2.2). • The assumption (A2) is a natural structural assumption in order to prove an entropy entropydissipation estimate like stated above. In fact, for general systems featuring boundary equilibria, the behaviour near a boundary equilibrium is unclear and able to prevent global exponential decay to an asymptotically stable equilibrium as can be seen in example systems. See Remark 2.2 for an example of a system having a boundary equilibrium. It is also remarked that there is a large class of systems possessing no boundary equilibria. See e.g. [CF06] for necessary condition to determine such systems. Lemma 2.2 (Uniqueness of detailed balance equilibrium). [GGH96, Lemma 3.4] If the system (2.2) satisfies (A1), then (2.2) has a unique detailed balance equilibrium. We recall that the assumption (A1) allows (without loss of generality) to rescale kr,f = kr,b = kr

for all

r = 1, . . . , R.

Thus, we define the entropy functional E(c) =

I Z X i=1

Ω

(ci log ci − ci + 1)dx,


9

which decays monotonically in time according to the following entropy-dissipation functional Z I Z R X X r r r r d |∇ci |2 dx + D(c) = − E(c) = di (x) kr (cα − cβ )(log cα − log cβ )dx ≥ 0. dt c i Ω r=1 i=1 Ω

(2.9)

Lemma 2.3 (L1 -bounds). Assume that the initial data c0 are nonnegative and satisfies E(c0 ) < +∞. Then, kci (t)kL1 (Ω) ≤ K := 2(E(c0 ) + I) ∀t > 0, ∀i = 1, 2, . . . , I. Proof. Integrating (2.9) over (0, t) leads to E(c(t)) ≤ E(c0 ) or equivalently I Z X ∀t > 0. (ci (x, t) log ci (x, t) − ci (x, t) + 1) dx ≤ E(c0 ) i=1

Ω

√ 2 By using the elementary inequalities x log x − x + 1 ≥ ( x − 1) ≥ 21 x − 1 for all x ≥ 0, we get I Z 1X ci (x, t) dx ≤ E(c0 ) + I. 2 i=1 Ω This, combined with the nonnegativity of solutions, completes the proof of the Lemma.

The following Csisz´ ar-Kullback-Pinsker type inequality shows that the convergence of equilibrium in L1 (Ω) of solutions follows from the convergence of the relative entropy E(c) − E(c∞ ) to zero. For a generalised Csisz´ ar-Kullback-Pinsker inequality, we refer to [AMTU01]. Here, we give an elementary proof using only the natural bound inheriting from Lemma 2.3. Lemma 2.4 (Csisz´ ar-Kullback-Pinsker type inequality). For all c ∈ L1 (Ω; [0, +∞)I ) such that Q c = Q c∞ and ci ≤ K for all i = 1, 2, . . . , I with some K > 0, we have E(c) − E(c∞ ) ≥ CCKP

I X

kci − ci,∞ k2L1 (Ω)

i=1

where the constant CCKP depends only on the domain Ω and the constant K. Proof. By recalling |Ω| = 1 and using the additivity of the relative entropy, we have I I Z X X ci ci ci log − ci + ci,∞ . E(c) − E(c∞ ) = ci log dx + ci ci,∞ i=1 i=1 Ω

(2.10)

By using the classical Csisz´ ar-Kullback-Pinsker inequality, we have Z ci (2.11) ci log dx ≥ C0 kci − ci k2L1 (Ω) c i Ω for all i = 1, 2, . . . , I, where the constant C0 depends only the domain Ω. On the other hand, by √ on √ applying the elementary inequality x log(x/y) − x + y ≥ ( x − y)2 , we obtain X I I I X 2 X √ ci (ci − ci,∞ )2 √ E(c) − E(c∞ ) = ci log − ci + ci,∞ ≥ ci − ci,∞ = 2 √ √ ci,∞ ci + ci,∞ i=1 i=1 i=1 (2.12) I 1 X 2 (ci − ci,∞ ) , ≥ 4K i=1 where K is given in Lemma 2.3. By combining (2.10)–(2.12), we obtain E(c) − E(c∞ ) ≥ C0

I X

kci − ci k2L1 (Ω) +

i=1

≥ min{C0 ; 1/4K}

I X

I 1 X (ci − ci,∞ )2 4K i=1

kci − ci k2L1 (Ω) + kci − ci,∞ k2L1 (Ω)

i=1 I X 1 min{C0 ; 1/4K} kci − ci,∞ k2L1 (Ω) , 2 i=1 1 which is the desired inequality with CCKP = 12 min C0 ; 4K .

≥

The following entropy entropy-dissipation estimate was established in [MHM15].

10


Theorem 2.5. [MHM15] Assume that (2.2) satisfies the assumption (A1) and (A2). For a given fixed positive initial mass vector M ∈ Rm + , there exists a positive constant λ > 0 such that D(c) ≥ λ(E(c) − E(c∞ )) 1

I

for all c ∈ L (Ω; [0, +∞) ) satisfying Q c = M, where c∞ is the detailed balance equilibrium of (2.2) corresponding to M. We emphasise that, though this Theorem gives the existence of λ > 0, it seems difficult to extract an explicit estimate of λ except in some special cases, e.g. a quadratic system arising from the reaction 2X Y . The main reason is that the proof used in [MHM15] is crucially based on a convexification argument, which appears very hard (if not impossible) to make explicit for general systems. In this paper, we propose a constructive method to prove an EED estimate. The method applies elementary estimates and has the advantage of a better computability of the rates and constants of convergence to equilibrium based on the structure of the conservation laws. Before detailing our approach, let us remark about the assumption (A2) on the absence of boundary equilibria. Remark 2.2 (Boundary equilibrium). The validity of Theorem 2.5 fails if the system (2.2) has a boundary equilibrium. For example, for the single reversible reaction 2A A + B with normalised reaction rate constants kf = kb = 1, we have the following system  2 x ∈ Ω, t > 0,  at − div(da (x)∇a) = −a + ab,  b − div(d (x)∇b) = a2 − ab, x ∈ Ω, t > 0, t b ∇a · ν = ∇b · ν = 0, x ∈ ∂Ω, t > 0,    a(x, 0) = a0 (x), b(x, 0) = b0 (x), x ∈ Ω. This system has one mass conservation law Z Z (a(x, t) + b(x, t))dx = (a0 (x) + b0 (x))dx =: M > 0 Ω

∀t > 0.

Ω

It is easy to see that the system possesses a positive detailed balance equilibrium (a∞ , b∞ ) = a boundary equilibrium (a∗∞ , b∗∞ ) = (0, M). Moreover, we have the entropy functional Z Z E(a, b) = (a log a − a + 1)dx + (b log b − b + 1)dx Ω

M M 2 , 2

and

Ω

and the entropy-dissipation functional Z Z Z |∇a|2 |∇b|2 D(a, b) = da (x) dx + db (x) dx + a(a − b)(log a − log b)dx. a b Ω Ω Ω By defining Z = {(a, b) ∈ R2+ : a + b = M}, we can easily compute that lim

∗ Z3(a,b)→(a∗ ∞ ,b∞ )

D(a, b) = 0

and lim

(E(a, b) − E(a∞ , b∞ )) = M log 2 > 0.

∗ Z3(a,b)→(a∗ ∞ ,b∞ )

As a consequence, there does not exist a constant λ > 0 such that

for all functions a, b : Ω → R+

D(a, b) ≥ λ(E(a, b) − E(a∞ , b∞ )) R satisfying Ω (a(x) + b(x))dx = M.

So, in general, if (2.2) has a boundary equilibrium then we cannot expect global exponential convergence to equilibrium. We can only expect exponential convergence of trajectories, which are uniformly bounded away from the boundary equilibrium. Interestingly, it is conjectured in the case of ODE reaction systems that even if the system possesses boundary equilibria, a trajectory starting from a positive initial state will always converge to the unique positive equilibrium as time goes to infinity. This conjecture is given the name Global Attractor Conjecture in [CDSS09] and is considered to be one of the most important open problems in the theory of chemical reaction networks. A recent proposed proof of the conjecture in ODE setting is under verification [Cra]. For PDE systems featuring boundary equilibria, we refer the reader to the recent paper [DFT].


11

2.2. A constructive method to prove the EED estimate. Though the Theorem 2.5 provides the existence of λ > 0 satisfying the entropy entropy-dissipation estimate, it does not seem to give an explicit estimates for λ when the reaction network has more than two substances, for example, αA + βB γC

or

A1 + A2 A3 + A4 .

Inspired by the works [DF08, DF14, FLT, FL16], we propose a general approach to prove an entropy entropy-dissipation estimate based on the mass conservation laws of a given general reaction-diffusion system of the form (1.1), which allows explicit estimates of the rates and constants of convergence. By recalling the crucial EED estimate D(c) ≥ λ(E(c) − E(c∞ )),

(2.13)

we point out that the right hand side is zero if and only if c ≡ c∞ , while the left hand side is zero for r r all constant states c∗ ∈ (0, +∞)I satisfying (c∗ )α = (c∗ )β , ∀r = 1, 2, . . . , R and that c∗ identifies with c∞ if and only if Q c∗ = M. Hence, the EED estimate (2.13) has to crucially take into account all the conservations laws of the system. The following notations and elementary inequalities are used in our proof: L (Ω)-norm: R For the rest of this paper, we will denote by k · k the usual norm of L2 (Ω): kf k2 = Ω |f (x)|2 dx. Spatial averages and square-root abbreviation: For a function f : Ω → R, the spatial average is denoted by (recall |Ω| = 1) Z f= f (x) dx. 2

Ω

Moreover, for a quantity denoted by small letters, we introduce the short hand notation of the same uppercase letter as its square root, e.g. √ √ Ci = ci , and Ci,∞ = ci,∞ . Additivity of Entropy: see e.g. [DF08, DF14], [MHM15, Lemma 2.3] E(c) − E(c∞ ) = (E(c) − E(c)) + (E(c) − E(c∞ )) I Z I X X ci ci = ci log dx + ci log − ci + ci,∞ . ci ci,∞ i=1 Ω i=1

(2.14)

The Poincaré inequality: For all f ∈ H 1 (Ω), there exists CP (Ω) > 0 depending only on Ω such that k∇f k2 ≥ CP kf − f k2 . The Logarithmic Sobolev inequality: Assume that 0 < dmin ≤ d(x) ≤ dmax < +∞ for all x ∈ Ω. Then, there exists CLSI (dmin ) > 0 depending on d(x) such that Z Z |∇f |2 f d(x) dx ≥ CLSI (dmin ) f log dx. (2.15) f f Ω Ω √ 2 √ An elementary inequality: (a − b)(log a − log b) ≥ 4 a − b . An elementary function: Consider Φ : [0, +∞) → [0, +∞) defined as (and continuously extended at z = 0, 1) Φ(z) =

z log z − z + 1 . √ 2 ( z − 1)

(2.16)

Then, Φ is increasing with lim Φ(z) = 1 and lim Φ(z) = 2. z→0

z→1

Remark 2.3 (Explicit constants). We remark that the rates obtained by our approach are explicit, yet not optimal. Therefore, for the sake of readability, we shall denote at several places by Ki explicit constants in the sense that Ki can be estimated explicitly, but we don’t give unnecessary long expression of Ki . The method of proving the EED estimate (2.13) consists of four steps designed as a chain of estimates, which allows to enter the conservation laws in a final step. Among the four steps, Step 1, Step 2. and Step 3. can be proved for general systems since their proofs do not rely on the particular structure of the conservation laws. In Step 4, which crucially uses the mass conservation laws defined in (2.8), an explicit constructive proof can be done for a given system (see the applications in Section 3 and Section

12


4) but for a general system it is unclear how to formalise Step 4 since the choice of the matrix Q is not unique in the general case. Nevertheless, we will see for the example systems in Section 3 and Section 4 that once the conservation laws are explicitly known, we can detail the proof of Step 4 and thus complete in this sense the proof of (2.13). Step 1 (Use of the Logarithmic Sobolev Inequality): The idea of this step is to divide the relative entropy E(c) − E(c∞ ) into two parts by using the additivity properties of the entropy (2.14): E(c) − E(c∞ ) = (E(c) − E(c)) + (E(c) − E(c∞ )) I Z I X X ci ci = ci log − ci + ci,∞ , ci log dx + ci ci,∞ i=1 Ω i=1

(2.17)

where the first part is controlled by the diffusion dissipation via the Logarithmic Sobolev Inequality, i.e. Z Z |∇ci |2 ci di (x) dx ≥ CLSI (di,min ) ∀i = 1, . . . , I ci log dx, ci ci Ω Ω and the second term on the right hand side of (2.17) depends only on spatial averages of concentrations, which have the two advantages of obeying the conservation laws and of satisfying the natural uniform a-priori bounds of Lemma 2.3. In particular, the Logarithmic Sobolev Inequality allows to estimate 1 1 D(c) ≥ min{CLSI (d1,min ), . . . , CLSI (dI,min )}(E(c) − E(c)). 2 2 and it remains to prove that 1 D(c) ≥ K1 (E(c) − E(c∞ )) 2

(2.18)

for an explicit constant K1 . Step 2 (Transformation into quadratic terms for the square roots of the concentrations): To prove (2.18), we first estimate D(c) below and then E(c)−E(c∞ ) above in terms of L2 -distance of the square roots Ci of the concentrations ci . The associated quadratic forms are significantly easier to handle than the logarithmic terms. For D(c) we estimate I

1 1X D(c) = 2 2 i=1 ≥2

I X

R

Z di (x) Ω

|∇ci |2 1X kr dx + ci 2 r=1 2

di,min k∇Ci k + 2

R X

Z

r

r

r

r

(cα − cβ )(log cα − log cβ )dx

Ω

(2.19)

r

r 2

kr Cα − Cβ ,

r=1

i=1

√ > where we recall the lower bounds di (x) ≥ di,min , the abbreviation √ C2i = ci , C = (C1 , C2 , . . . , CI ) √ and the elementary inequality (a − b)(log a − log b) ≥ 4( a − b) . For the second terms on the right hand side of (2.17) we use the function Φ in (2.16) to estimate X I I X 2 √ ci ci √ E(c) − E(c∞ ) = ci log − ci + ci,∞ = Φ ci − ci,∞ c c i,∞ i,∞ i=1 i=1 (2.20) q I 2 X 2 ≤ K2 Ci − Ci,∞ i=1

with K Φ , i=1,...,I ci,∞

K2 = max

where we used Lemma 2.3 to estimate ci ≤ K for all i = 1, . . . , I.

(2.21)


13

Thus, after the Steps 1 and 2, in particular from (2.19) and (2.20), it follows that in order to prove (2.18), it remains to find an explicit constant K1 > 0 such that 2 I R I q

2

r X X X

α 2 βr 2 (2.22) Ci − Ci,∞ . 2 di,min k∇Ci k + 2 kr C − C ≥ K1 K2 r=1

i=1

i=1

Step 3 (Control of reaction dissipation term by a reaction dissipation term for averages): The inequality (2.22) exhibits still the same fundamental structure as the desired inequality (2.18) in the sense that all the conservation law are required in order to see that the left hand side of (2.22) is zero only at the equilibrium Ci,∞ , which is also the unique constant state for which the right hand side of (2.22) is zero. Step 3 continues to estimate (2.22) in a way, which makes it possible in Step 4 below to use the conservation laws explicitly. The two terms on left hand side of (2.22) represents lower bounds for the entropy-dissipation caused by all the diffusion and reaction processes of the considered system. In order to be able to use the constrains provided by the conservation laws, we shall bound the reaction dissipation term below by a reaction dissipation term for spatially averaged concentrations. More precisely, by denoting C = (C1 , . . . , CI )> , we have Lemma 2.6 (Reaction dissipation terms for averaged concentrations). There exists an explicit constant K3 > 0 such that 2

I X

di,min k∇Ci k2 + 2

i=1

R X

r

r

kr kCα − Cβ k2 ≥ K3

r=1

I X i=1

k∇Ci k2 +

R X

αr

C

−C

βr

2

.

(2.23)

r=1

We postpone a proof of this Lemma to the end of this section for the sake of a continued presentation the main ideas of our strategy. It’s worth noticing that comparing to related proof given in [DF08], in which the considered quadratic nonlinearities allowed to exploit certain L2 orthogonality structures, Lemma 2.6 is more complicated due to the arbitrary order of the considered nonlinearities. In the proof of Lemma 2.6, we shall introduce some new ideas, which are motivated by [FL16] and consist of a domain decomposition to overcome the difficulties caused by the nonlinearities. This idea is also applicable to volume-surface reaction-diffusion systems, see [FLT]. By combining (2.22) and (2.23), the remaining open problem is to find an explicit K1 > 0 satisfying 2 I R I q X X K2 X αr βr 2 2 2 k∇Ci k + C −C ≥ K1 Ci − Ci,∞ (2.24) K3 i=1 r=1 i=1 with K2 is defined in (2.21) and K3 is in (2.23). Note that (2.24) is close to a finite dimensional inequality for the averaged square root concentrations Ci , if one would imagine to use the gradient term on the left hand side to replace somehow the Ci2 terms on the right hand side by terms depending on Ci . However, the averages Ci are not suitable for expressing the conservations law and Step 4 will instead apply a different approach. Step 4 (Conservation laws and deviations of averaged concentrations from equilibrium): Before continuing, we remark that while the previous three steps can be proved in the general case without knowing details of the structure of the conservation laws, Step 4 is rather a proof of concept how to proceed to complete the proof of the EED estimate for a specified model, whose conservation laws are explicit given (see Lemmas 3.1 and 4.1 for examples of two specific models). To prove (2.24), we use the ansatz 2 Ci2 = Ci,∞ (1 + µi )2

for all

i = 1, . . . , I

(2.25)

or equivalently C2 = C2∞ (1 + µ)2 with 1 = (1, 1, . . . , 1)> ∈ RI and µ = (µ1 , . . . , µI )> , where the non-negativity of the concentrations implies that µ ≥ −1. By recalling that C2 = c and C2∞ = c∞ and Q c = Q c∞ = M, we

14


have moreover the following algebraic constrains between the µ1 , . . . , µI : Q C2∞ (1 + µ)2 = Q C2∞ or equivalently Q C2∞ µ(µ + 2) = 0. By denoting δi (x) = Ci (x) − C i the deviation of Ci (x) to its average and by using (2.25), it 2 follows from kδi k2 = Ci2 − Ci that q kδi k2 Ci = Ci2 − q =: Ci,∞ (1 + µi ) − kδi k2 R(Ci ), (2.26) Ci2 + Ci q −1 Ci2 + Ci for all i = 1, . . . , I. We observe that R(Ci ) becomes where we denote R(Ci ) := 2

unbounded when Ci2 ≥ Ci approaches zero. The possible occurrence of such degenerate states (with arbitrarily little mass Ci2 of the species Ai being present) prevents a uniform use of the ansatz (2.26). Therefore, we have to distinguish two cases where Ci2 is either ”big”, say Ci2 ≥ ε2 2 for a ε2 to be chosen smaller than the equilibrium mass, e.g. ε2 = Ci,∞ /10, or ”small” Ci2 ≤ ε2 . We remark that a good value for the constant ε > 0 can be explicitly computed in specific models (see (3.18) in Section 3 or (4.9) in Section 4). Case 1. When Ci2 ≥ ε2 for all i = 1, . . . , I. In this case, we have −1 q 1 Ci2 + Ci ≤ , R(Ci ) = ε

∀i = 1, 2, . . . , I.

Thus, we can estimate the left hand side of (2.24) for a θ ∈ (0, 1) to be chosen (and by using Poincaré’s inequality and kδi k2 ≤ ci ≤ K with Lemma 2.3) as follows " I R Y I X X α r 2 Ci,∞ (1 + µi ) − kδi k2 R(Ci ) i LSH of (2.24) ≥ CP kδi k + θ r=1 i=1

i=1

−

I Y

β r Ci,∞ (1 + µi ) − kδi k R(Ci ) i

#2

2

i=1

≥ CP

I X

kδi k2 + θ

i=1

R h X

r

r

r

α β Cα − Cβ ∞ (1 + µ) ∞ (1 + µ)

r

i2

− θ C(ε, K)

r=1

kδi k2

i=1

R h n r o2 X i r r 2 ≥ θ min Cα (1 + µ)α − (1 + µ)β , ∞ r=1,...,R

I X

(2.27)

r=1 r

r

β where we have used the detailed balance condition, Cα ∞ = C∞ and chosen θ ∈ (0, 1) such that θC(ε, K) ≤ CP where C(ε, K) is a constant explicitly depends on ε and K. On the other hand, by applying the ansatz (2.25) to the right hand side of (2.24), it follows directly that 2 I q I X K2 K2 X 2 2 Ci − Ci,∞ ≤ K1 max {Ci,∞ } µ2i . (2.28) K1 K3 i=1 K3 i=1,...,I i=1

Thus, by combining (2.27) and (2.28), it is sufficient for the desired inequality (2.24) to find a constant K1 > 0 such that the following finite dimensional inequality holds under the constrains posed by the conservation laws, i.e. R h X r=1

r

(1 + µ)α − (1 + µ)β

r

i2

≥ K1

I 2 K2 maxi=1,...,I {Ci,∞ } X µ2i r 2 θK3 minr=1,...,R {Cα } ∞ i=1

(2.29)

with Q C2∞ µ(µ + 2) = 0. To prove (2.29) explicitly, we require the explicit form (or at least an explicit structure) of the mass conservation laws represented by Q, which is known for a specific model but unclear for general systems. In this article, we will present two specific systems, which allow


15

to give an explicit proof of (2.29): Lemma 3.3 for a single reversible reaction and in Lemma 4.3 for a chain of reversible reactions in which the conservation laws are explicitly known. Case 2. When Ci20 ≤ ε2 for some i0 ∈ {1, . . . , I}. We first bound the right hand side of (2.24) above by using the boundedness of averaged concentrations ci ≤ K in Lemma 2.3: I 4IKK2 2K2 X 2 2 ≤ K1 Ci + Ci,∞ RHS of (2.24) ≤ K1 . (2.30) K3 i=1 K3 Then, we bound the left hand side of (2.24) below by a positive constant. This quantifies the observation that degenerate states with Ci20 ≤ ε2 for ε2 sufficiently smaller than the equilibrium mass, say Ci20 ,∞ /10, can not be close to equilibrium and thus dissipate entropy with a strictly positive rate, or in other words the entropy-dissipation is bounded below by a strictly positive constant. In order to do so, we have to consider another two subcases due to the role of the diffusion. (i) (Diffusion is dominant.) If kδi∗ k2 ≥ C(ε, i0 ) for some i∗ ∈ {1, . . . , I}, where C(ε, i0 ) is an explicit constant in terms of ε and i0 to be chosen below, see (2.33). Then, the left hand side of (2.24) is bounded below by LSH of (2.24) ≥ CP

I X

kδi k2 ≥ CP C(ε, i0 ).

(2.31)

i=1

Hence, (2.24) follows from (2.30) and (2.31) by choosing K1 > 0 sufficiently small such that at least K3 CP C(ε, i0 ) . K1 ≤ 4IKK2 (ii) (Diffusion is insufficient.) If kδi k2 ≤ C(ε, i0 ) for all i = 1, . . . , I, then we can estimate 2

Ci = Ci2 − kδi k2 ≥ Ci2 − C(ε, i0 ),

∀i = 1, 2, . . . , I.

(2.32)

Moreover, we recall that Ci20 ≤ ε2 , i.e. that we consider a state with sufficiently less mass contained in Ci20 as in its equilibrium value Ci20 ,∞ . Thus, via a contradiction argument, the mass conservation laws Q C2 = Q C2∞ imply that there exists at least one concentration containing more mass than its equilibrium value, i.e. Cj2∗ ≥ Cj2∗ ,∞ . Thus, by choosing C(ε, i0 ) such that C(ε, i0 ) ≤

Cj2∗ ,∞ , 2

(2.33)

we obtain from (2.32) that Cj2∗ ,∞ . (2.34) 2 What we aim to do next is to exploit (several) bounds like (2.34) to show that the reaction dissipation is bounded below by a strictly positive constant, that is R X αr βr 2 C −C ≥ K4 (ε, i0 , j∗ , M). (2.35) 2

C j∗ ≥

r=1

This inequality depends on the explicit structure of the structure of {β r − αr : r = 1, . . . , R} and thus remains as an open problem in the general case. Explicit estimates of the form (2.35) are presented in this paper for special cases: estimate (3.19) in the proof of Theorem 1.1 and estimates (4.19) in the proof of Theorem 1.2. Once (2.35) is obtained, (2.24) follows from (2.30) choosing K1 sufficiently small to also satisfy K1 ≤

K3 K4 (ε, i0 , j∗ , M) . 4IKK2

Remark 2.4. For convenience of the reader, we recall here that in any specific model, it remains to complete the proof of Step 4 by • proving explicitly the finite dimensional inequality (2.29); and • proving explicitly the lower bound of the reaction dissipation (2.35).

16


For the rest of this section, we give a proof of Lemma 2.6 stated in Step 3. Proof of Lemma 2.6. We first prove that I X

di,min k∇Ci k2 + 2

R X

R

r X r 2 αr βr 2

C −C kr Cα − Cβ ≥ κ

(2.36)

r=1

r=1

i=1

for an explicit constant κ > 0. Then, (2.23) follows by choosing K3 = min min {di,min }; κ . i=1,...,I

The proof of (2.36) introduces pointwise deviations of the concentrations around their spatial averages, which are as follows: for all i = 1, 2, . . . , I, we define for x ∈ Ω.

δi (x) = Ci (x) − C i ,

√ Thanks √ to the non-negativity of Ci and Lemma 2.3, we have that δi ∈ [− K; +∞). Fixing a constant L > K > 0, we can decompose Ω as Ω = S ∪ S⊥, where S = {x ∈ Ω : |δi (x)| ≤ L,

∀i = 1, 2, . . . , I}.

⊥

We will prove (2.36) on both S and S . On S we estimate for a γ ∈ (0, 2) by using Taylor expansion (i.e. αri PI QI αr = C + i=1 Ri (C, αir , δ)δi with bounded remainders |Ri (C, αir , δ)| ≤ C(K, αr , L)) i=1 C i + δi and Young’s inequality that

γ

R X

r

r 2

kr Cα − Cβ 2

r=1

=γ

L (S)

R X r=1

≥

I

2 I

Y αri Y βir

kr C i + δi C i + δi −

2

i=1

i=1

R X

1

αr γ min {kr }

C 2 r=1,...,R r=1

L (S)

β r 2 −C 2

− γ C(L)

L (S)

I X

kδi k2L2 (S) , (2.37)

i=1

where C(L) is an explicit constant which does not depend on S. On the other hand, by using the Poincaré inequality, k∇f k2 ≥ CP kf − f k2 ≥ CP kf − f k2L2 (S) , we have I I I X X 1 1 1X di,min k∇Ci k2 ≥ CP min {di,min } kδi k2 ≥ CP min {di,min } kδi k2L2 (S) . i=1,...,I i=1,...,I 2 i=1 2 2 i=1 i=1

(2.38)

From (2.37) and (2.38), if we choose γ ∈ (0, 2) such that 2γC(L) ≤ CP min {di,min }, then we have i=1,...,I

I R R

r

X X r 2 1X 1 β r 2

αr di,min k∇Ci k2 + 2 kr Cα − Cβ 2 ≥ γ min {kr }

C − C 2 . 2 i=1 2 r=1,...,R L (S) L (S) r=1 r=1

(2.39)

To estimate (2.36) on S ⊥ , we note that by the definition of S ⊥ and L, we have S ⊥ = {x ∈ Ω :

δi (x) > L for some i = 1, 2 . . . , I}.

Hence, |S ⊥ | =

I X

|{x ∈ Ω : δi (x) > L}| =

i=1

1 ≤ 2 L

I X {x ∈ Ω : δi2 (x) > L2 } i=1

I X i=1

I

2

kδi k ≤

L2 CP

X 1 di,min k∇Ci k2 . min {di,min } i=1

i=1,...,I

(2.40)


By making use of the following a priori bounds Ci ≤ right hand side of (2.36) in S ⊥ as follows R

X β r 2

αr

C − C 2

L (S ⊥ )

r=1

√ ≤ C( K) S ⊥ ≤

≤

I 1X

2

L2 CP

√ I X C( K) di,min k∇Ci k2 min {di,min } i=1

(use (2.40))

i=1,...,I

i=1

if we choose L to be big enough, e.g. L ≥ (2.36) with κ =

q √ Ci2 ≤ K from Lemma 2.3, we can estimate the

di,min k∇Ci k2 , 2

1 2

17

CP

√ 2C( K) min {di,min } .

(2.41) By combining (2.39) and (2.41) we obtain

i=1,...,I

min{1, γ min {kr }}.

r=1,...,R

3. A single reversible reaction - Proof of Theorem 1.1 In this section, we will prove Theorem 1.1 and in particular the entropy entropy-dissipation estimate (1.11) with an explicit constant λ, which yields convergence to equilibrium for a single reversible reaction of the form α1 A1 + α2 A2 + . . . + αI AI β1 B1 + β2 B2 + . . . + βJ BJ with stoichiometric coefficients αi , βj ≥ 1 for i = 1, . . . , I and j = 1, . . . , J for any I, J ≥ 1. For the sake of convenience and w.l.o.g. we rescale the forward and backward reaction rate constants to kf = kb = 1. An explicit entropy entropy-dissipation estimate was left open in [MHM15] whenever I + J ≥ 3. The reaction is assumed to take place in reaction vessel, i.e. in a bounded domain Ω ⊂ Rn , n ≥ 1 with sufficiently smooth boundary ∂Ω (e.g. ∂Ω ∈ C 2+ for some > 0). The mass action reaction-diffusion system reads as  ∂t ai − div(da,i (x)∇ai ) = −αi (aα − bβ ), in Ω × R+ , i = 1, 2, . . . , I,    ∂ b − div(d (x)∇b ) = −β (aα − bβ ), in Ω × R+ , j = 1, 2, . . . , J, t j b,j j j (3.1)  ∇a · ν = ∇b · ν = 0, on ∂Ω × R+ , i = 1, . . . , I, j = 1, . . . , J, i j    a(x, 0) = a0 (x), b(x, 0) = b0 (x), in Ω, where da,i (x), db,j (x) are diffusion coefficients satisfying 0 < dmin ≤ da,i (x), db,j (x) ≤ dmax

∀x ∈ Ω, i = 1, . . . , I, j = 1, . . . , J,

a = (a1 , a2 , . . . , aI ), b = (b1 , b2 , . . . , bJ ) are the vectors of left- and right-hand-side concentrations, α = (α1 , α2 , . . . , αI ) ∈ [1, +∞)I and β = (β1 , β2 , . . . , βJ ) ∈ [1, +∞)J are the corresponding vectors of stoichiometric coefficients and we recall the notation aα =

I Y i=1

i aα i

and

bβ =

J Y

β

bj j .

j=1

The aim of this section is to apply the method proposed in Section 2 to prove the entropy entropydissipation estimate (1.11), which implies explicit convergence to equilibrium for the system (3.1). First, we shall specify the mass conservation laws for (3.1), which are essential to our strategy. Then, (3.1) is shown to satisfy the assumptions (A1) and (A2), that is (3.1) satisfies the detailed balance condition and has no boundary equilibrium. Finally, we prove the main result of this section Theorem 1.1. Lemma 3.1 (Mass conservation laws). The system (3.1) obeys I + J − 1 linear independent mass conservation laws. With respect to the general notation of conservation laws (2.7), the matrix Q can be chosen as Q = [v1 , . . . , vJ , w2 , . . . , wI ]> ∈ R(I+J−1)×(I+J) where the zero left-eigenvectors vj and wi are defined by 1 1 1 1 vj = , 0, . . . , 0, , 0, . . . , 0 and wi = 0, . . . , 0, , 0, . . . , 0, , 0, . . . , 0 , α1 βj αi β1 | {z } | {z } i I+j | {z } I+1

for 1 ≤ j ≤ J and 2 ≤ i ≤ I.

(3.2)

18


Proof. From (3.1), by dividing the equation of ai by αi and the equation of bj by βj , summation and integration over Ω yields, thanks to the homogeneous Neumann boundary condition, Z d ai (x, t) bj (x, t) + dx = 0 ∀t > 0. dt Ω αi βj R a (x) b (x) Hence, after introducing the nonnegative partial masses Mi,j := Ω i,0αi + j,0βj dx, we observe that system (3.1) obeys the following IJ mass conservation laws ai (t) bj (t) + = Mi,j , ∀t > 0, ∀i = 1, . . . , I, ∀j = 1, . . . , J, αi βj R where we recall the notation ai = Ω ai (x)dx. The following I + J − 1 conservations laws a1 bj + = M1,j α1 βj

and

ai b1 + = Mi,1 αi β1

for

j = 1, . . . , J,

(3.3)

i = 2, . . . , I.

are linear independent since the corresponding zero left-eigenvectors vj and wi as given in (3.2) are linear independent by construction. Moreover, all other conservation laws can be expressed in terms of these I + J − 1 conservation laws due to ai bj b1 a1 + = Mi,1 − + M1,j − = Mi,1 + M1,j − M1,1 . αi βj β1 α1 Remark 3.1. It follows from the Lemma 3.1 that the initial mass vector M is uniquely determined by prescribing its I + J − 1 coordinates M1,j with 1 ≤ j ≤ J and Mi,1 . Therefore, from now on, by calling the initial mass vector M fixed, we mean that those coordinates are prescribed. Remark 3.2. Another useful set of conservation laws follows from dividing the equation for ai by αi and the equation for ak by αk for 1 ≤ i 6= k ≤ I: Z Z ai (t, x) ak (t, x) ai,0 (x) ak,0 (x) − − dx = Ni,k := dx, ∀t ≥ 0, 1 ≤ i 6= k ≤ I, αi αk αi αk Ω Ω It’s also useful to observe that Ni,k = Mi,j − Mk,j ,

∀1 ≤ j ≤ J.

(3.4)

Lemma 3.2 (Unique constant positive equilibrium). For any fixed positive initial mass vector M, the system (3.1) possesses a unique positive equilibrium (a∞ , b∞ ) ∈ (0, +∞)I+J solving   ai,∞ + bj,∞ = M , i = 1, . . . , I, j = 1, . . . , J, i,j αi βj (3.5)  α a∞ = bβ ∞. Consequently, system (3.1) satisfies the assumptions (A1) and (A2). Proof. Without loss of generality, we may assume that a1,∞ ai,∞ = min . i=1,2,...,I α1 αi Then, from (3.5) and (3.4), we have for all i > 1 a1,∞ ai,∞ − = Mi,k − M1,k = Ni,1 ≥ 0 αi α1 and thus αi I I Y Y αi α1 i aα a =: f (a1,∞ ), = a α N + i i,1 1,∞ 1,∞ i,∞ α1 i=1 i=2

(3.6)

which is a strictly monotone increasing function in a1,∞ . From (3.5), we deduce similarly that bj,∞ = β βj M1,j − αj1 a1,∞ ≥ 0 and thus βj J J Y Y βj βj bj,∞ = βj M1,j − a1,∞ =: g(a1,∞ ), α1 j=1 j=1 which is a strictly monotone decreasing function in a1,∞ . From the definition, it is obvious that f (0) = 0 and g(0) > 0. On the other hand, we have a1,∞ ≤ minj=1,...,J {α1 M1,j } =: a1,max and consequently f (a1,max ) > 0 and g(a1,max ) = 0. Thus the equation


19

f (a1,∞ ) = g(a1,∞ ) has a unique positive solution a1,∞ ∈ (0, a1,max ). Hence, there exists a unique positive equilibrium (a∞ , b∞ ). It is straightforward that the assumption (A1) holds. The assumption (A2) can be easily verified by assuming ai0 ,∞ = 0 for some 1 ≤ i0 ≤ I, which implies aα ∞ = 0. Then, by (3.5), bj,∞ = Mi0 ,j > 0 for j = 1, 2, . . . , J, and thus bβ > 0, which is a contradiction. We conclude ai,∞ > 0 for all i = 1, 2, . . . , I and ∞ similarly, bj,∞ > 0 for all j = 1, 2, . . . , J. Therefore, the system (3.1) has no boundary equilibrium. The entropy functional for system (3.1) writes as E(a, b) =

I Z X i=1

(ai log ai − ai + 1)dx +

Ω

J Z X

(bj log bj − bj + 1)dx

Ω

j=1

and the entropy-dissipation writes as Z J Z I Z X X |∇bj |2 aα |∇ai |2 dx + dx + (aα − bβ ) log β dx. db,j (x) D(a, b) = da,i (x) ai bj b Ω j=1 Ω i=1 Ω Proof of Theorem 1.1. We follow the strategy in Section 2 to prove this Theorem. Note that due to the explicitly specified mass conservation laws of Lemma 3.1, we can completely detail the proof of Step 4. For convenience of the reader, we shall briefly recall the main steps in terms of the variables a and b. Step 1. Thanks to the additivity of the relative entropy, we have E(a, b) − E(a∞ , b∞ ) = (E(a, b) − E(a, b)) + (E(a, b) − E(a∞ , b∞ )) By using the Logarithmic Sobolev Inequality (2.15), we get 1 1 D(a, b) ≥ min {CLSI (da,i ), CLSI (db,j )} (E(a, b) − E(a, b)). 2 2 i,j and it is left to find K1 > 0 such that 1 (3.7) D(a, b) ≥ K1 (E(a, b) − E(a∞ , b∞ )). 2 √ √ √ √ √ Step 2. With 2∇ f = ∇f / f , (a − b)(log a − log b) ≥ 4( a − b)2 and the notation Ai = ai , Bj = p bj , A = (A1 , . . . , AI ) and B = (B1 , . . . , BJ ), we estimate the following lower and upper bounds: X J I X 1 2 2 k∇Bj k + 2kAα − B β k2 , D(a, b) ≥ 2dmin k∇Ai k + 2 i=1 j=1 E(a, b) − E(a∞ , b∞ ) ≤ K2

X I q

A2i

2 − Ai,∞

+

i=1

J q X

Bj2

2 − Bj,∞

,

j=1

for an explicit constant K2 > 0. Thus, (3.7) follows from proving the following estimate for an explicit constant K1 > 0: 2 X 2 X X I J I q J q X A2i −Ai,∞ + Bj2 − Bj,∞ 2dmin k∇Ai k2 + k∇Bj k2 +2kAα −B β k2 ≥ K1 K2 i=1

j=1

j=1

i=1

(3.8) Step 3. It follows from Lemma 2.6 that X X I J I J α X X β 2 . 2dmin k∇Ai k2 + k∇Bj k2 + 2kAα − B β k2 ≥ K3 k∇Ai k2 + k∇Bj k2 + A − B i=1

j=1

i=1

j=1

(3.9) for an explicit K3 > 0. Thus, (3.8) follows from (3.9) for a sufficiently small K1 > 0 provided that the following estimate holds !2 2 I J I q J q α X X X K2 X β 2 2 2 2 k∇Ai k + k∇Bj k + A − B ≥ K1 Ai − Ai,∞ + Bj2 − Bj,∞ . K3 i=1 i=1 j=1 j=1 (3.10)

20


Step 4. We introduce the ansatz A2i = A2i,∞ (1+µi )2 ,

2 (1+ξj )2 , Bj2 = Bj,∞

µi , ξj ∈ [−1, +∞),

∀i = 1, . . . , I, j = 1, . . . , J. (3.11)

By using the deviations δi (x) = Ai (x) − Ai

∀x ∈ Ω,

ηj (x) = Bj (x) − Bj

and

∀x ∈ Ω,

we have q kδi k2 Ai = A2i − q =: Ai,∞ (1 + µi ) − R(Ai )kδi k2 2 Ai + Ai

with R(Ai ) = q

1 A2i + Ai

and similarly 1 . with R(Bj ) = q Bj2 + Bj

Bj = Bj,∞ (1 + ξj ) − R(Bj )kηj k2 We now consider two cases:

Case 1. We first consider A2i ≥ ε2 and Bj2 ≥ ε2 for all i = 1, . . . , I, j = 1, . . . , J and remark that ε is set explicitly in (3.18) below. Moreover, the remainder terms R(Ai ) and R(Bj ) are bounded in terms I I Q Q 2β 2αi = Bj,∞j . of ε. Thus, we estimate with Ai,∞ i=1

i=1

α

A −B

β 2

=

Y I

Ai,∞ (1 + µi ) + Q(Ai )kδi k

2 αi

i=1

≥

I Y

−

J Y

2 βj

2

Bj,∞ (1 + ξj ) + Q(Bj )kηj k

j=1

i A2α i,∞

α

(1 + µ) − (1 + ξ)

β

2

X I J X 2 2 kδi k + kηj k − C(ε, K)

i=1

i=1

j=1

−1

Therefore, by choosing 0 < (1 − θ) ≤ CP C(ε, K) where CP is the Poincaré constant, we can estimate the left hand side of (3.10) as follows X I I J 2 α Y X β 2 α β 2 2 i A2α . ≥θ (1 − θ + θ) k∇Ai k + k∇Bj k + A − B i,∞ (1 + µ) − (1 + ξ) i=1

j=1

i=1

(3.12) On the other hand the right hand side of (3.10) is bounded by ! I J X X K1 K2 2 2 2 2 max{Ai,∞ , Bj,∞ } µi + ξj . K3 i,j i=1 j=1 Hence, (3.10) follows from (3.12) provided the following −1 Y X I I J 2 X K2 α β 2αi 2 2 2 2 (1 + µ) − (1 + ξ) ≥ K1 Ai,∞ µi + ξj . max{Ai,∞ , Bj,∞ } θK3 i,j i=1 i=1 j=1 for a suitable K1 > 0. This finite dimensional inequality follows from Lemma 3.3 below with Y −1 I 1 K2 2αi 2 2 A max{Ai,∞ , Bj,∞ } . (3.13) K1 ≤ max{I, J} i=1 i,∞ θK3 i,j Case 2. When A2i ≤ ε2 or Bj2 ≤ ε2 for some i = 1, . . . , I, j = 1, . . . , J. First, we observe that the right hand side of (3.10) is bounded above by I q 2 X 2 J q K1 K3 X K1 K2 A2i − Ai,∞ + Bj2 − Bj,∞ ≤ 4(I + J)K K2 K3 i=1 j=1

(3.14)

thanks to the natural bounds of ai ≤ K and bj ≤ K in Lemma 2.3. Next, without loss of generality, we can assume that A2i0 ≤ ε2 for some 1 ≤ i0 ≤ I. Since such degenerate states are far from equilibrium for ε2 sufficiently smaller than the equilibrium mass ai0 ,∞ , it is sufficient and now possible to show that the left hand side of (3.10) is bounded below by a positive constant. We will have to consider two subcases due to the amount of diffusion represented by the values of kδi k2 and kηj k2 :


21 2

2

(i) (Diffusion is dominant.) Consider that kδi∗ k2 ≥ αεi for some 1 ≤ i∗ ≤ I or kηj∗ k2 ≥ αεi 0 0 for some 1 ≤ j∗ ≤ J. Then, thanks to the Poincaré inequality, the left hand side of (3.10) obviously bounded below by X I J X CP ε2 2 2 LSH of (3.10) ≥ CP . (3.15) kδi k + kηj k ≥ αi0 i=1 j=1 By combining (3.15) and (3.14), we obtain (3.10) whenever K1 ≤

CP ε2 K3 . 4αi0 (I + J)KK2

(3.16)

2

(ii) (Diffusion is insufficient.) Consider that kδi k2 ≤ αεi for all i = 1, . . . , I and kηj k2 ≤ 0 all j = 1, . . . , J. Then, by using the mass conservation (3.3),

ε2 αi0

for

Bj2 A2i0 + = Mi0 ,j , αi0 βj we have with A2i0 ≤ ε2 A2 ε2 Bj2 = βj Mi0 ,j − i0 ≥ βj Mi0 ,j − , αi0 αi0 for all j = 1, . . . , J. Hence, for all j = 1, . . . , J, 2

Bj = Bj2 − kηj k2 ≥ βj Mi0 ,j −

βj + 1 2 ε . αi0

Then, we estimate the left hand side of (3.10) with an elementary inequality (a−b)2 ≥ 12 a2 −b2 , A2i0 ≤ ε2 and Ai Y I i=1

Ai

αi

−

2αi

J Y

≤ A2i

Bj

βj

αi

≤ (αi Mi,1 )αi for all i 6= i0 as follows:

2 ≥

j=1

J I 1 Y 2βj Y 2αi Bj − Ai 2 j=1 i=1

β J 1Y βj + 1 2 j ≥ βj Mi0 ,j − ε − ε2 2 j=1 αi0

I Y

(3.17) αi

(αi Mi,1 ) .

i=1,i6=i0

We now choose ε small enough such that both inequalities βj Mi0 ,j −

βj Mi0 ,j βj + 1 2 ε ≥ αi0 2

for all

j = 1, . . . , J

and β J 1 Y βj Mi0 ,j j − ε2 2 j=1 2

I Y

αi

(αi Mi,1 )

i=1,i6=i0

β J 1 Y βj Mi0 ,j j ≥ 4 j=1 2

hold, e.g. ε fulfills Y −1 Y β I J αi0 βj Mi0 ,j 1 βj Mi0 ,j j 2 αi ε ≤ min min ; (αi Mi,1 ) . 1≤j≤J 4(βj + 1) 4 2 j=1

(3.18)

i=1,i6=i0

It then follows from (3.17) that Y I

Ai

αi

i=1

−

J Y j=1

Bj

βj

2

β J 1 Y βj Mi0 ,j j ≥ . 4 j=1 2

(3.19)

Therefore, (3.10) follows from (3.14) and (3.19) provided K1 ≤

β J Y K3 βj Mi0 ,j j . 8KK2 (I + J) j=1 2

(3.20)

22


Altogether, from (3.13), (3.16) and (3.20), we obtain (3.10) and consequently the desired entropy entropy-dissipation estimate D(a, b) ≥ λ(E(a, b) − E(a∞ , b∞ )) with the explicit constant 1 λ = min min{CLSI (da,i ), CLSI (db,j )}; 2K1 . i,j 2 Thus, by applying this functional inequality to solutions of (3.1), a Gronwall argument implies directly exponential decay to equilibrium in relative entropy, from which the Ciszár-Kullback-Pinsker inequality yields to following exponential convergence to equilibrium: I X

kai (t) − ai,∞ k2L1 (Ω) +

i=1

J X

−1 (E(a0 , b0 ) − E(a∞ , b∞ ))e−λt . kbj (t) − bj,∞ k2L1 (Ω) ≤ CCKP

j=1

Thus, the proof of Theorem 1.1 is finished.

Lemma 3.3. Let µi and ηj be defined as in (3.11). Then, the following inequality holds Y I

(1 + µi )αi −

i=1

J Y

(1 + ξj )βj

2 ≥

j=1

X I J X 1 µ2i + ξj2 . max{I, J} i=1 j=1

(3.21)

Proof. The proof of (3.21) relies on relations between µi and ξj arising from mass conservation laws (3.3) 2 for all 1 ≤ i ≤ I and 1 ≤ j ≤ J. Hence, and (3.2). In particular, we have A2i + Bj2 = Mi,j = A2i,∞ + Bj,∞ by using (3.11), we obtain 2 A2i,∞ µi (µi + 2) + Bj,∞ ξj (ξj + 2) = 0,

for all

1 ≤ i ≤ I,

1 ≤ j ≤ J.

(3.22)

Since µi , ξj ∈ [−1, +∞), it follows from (3.22) that µi and ξj must have opposite signs. In particular, if µi ≥ 0 for some i = 1, 2, . . . , I, then ξj ≤ 0 for all j = 1, 2, . . . , J and in return µi ≥ 0 for all i = 1, 2, . . . , I. Therefore, we have only two cases to consider: (I) Let µi ≥ 0 and ξj ≤ 0 for all i = 1, 2, . . . , I and j = 1, 2, . . . , J, or (II) Let µi ≤ 0 and ξj ≥ 0 for all i = 1, 2, . . . , I and j = 1, 2, . . . , J. We will only prove case (I) since case (II) can be treated similarly. Now since µi ≥ 0 and −1 ≤ ξj ≤ 0, we can apply Jensen inequality for any 1 ≤ i∗ ≤ I and 1 ≤ j∗ ≤ J to estimate Y Y J I J I J Y Y Y Y I αi βj βj (1 + µi )αi − (1 + ξj ) (1 + µ ) (1 + ξ ) (1 + µ ) − (1 + ξ ) ≥ − ≥ i j i j i=1

j=1

i=1

j=1

i=1

j=1

≥ (1 + µi∗ ) − (1 + ξj∗ ) = µi∗ − ξj∗ ≥ 0. Hence, for all 1 ≤ i∗ ≤ I and 1 ≤ j∗ ≤ J Y 2 I J Y αi βj (1 + µi ) − (1 + ξj ) ≥ (µi∗ − ξj∗ )2 ≥ µ2i∗ + ξj2∗ i=1

j=1

and (3.21) follows immediately.

4. Reversible enzyme reactions - Proof of Theorem 1.2 In this section, we demonstrate the strategy in Section 2 for a chain of two reversible reactions modelling, for instance, reversible enzyme reactions. More precisely, we consider the system A1 + A2 A3 A4 + A5 .

(4.1)

In [BCD07] and [BP10], this reaction was studied in the context of performing a quasi-steady-stateapproximation, i.e. that the releasing reaction rate constant from A3 to A1 + A2 and from A3 to A4 + A5 are taken to diverge to infinitely. Here in contrast, we shall assume all the reaction constants to be one. As in the previous section, we assume the reaction to occur on a bounded domain Ω ⊂ Rn with smooth boundary ∂Ω and normalised volume |Ω| = 1. By applying the mass action law, the corresponding


reaction-diffusion system of (4.1) reads as  ∂t c1 − div(d1 (x)∇c1 ) = −c1 c2 + c3 ,      ∂t c2 − div(d2 (x)∇c2 ) = −c1 c2 + c3 ,    ∂t c3 − div(d3 (x)∇c3 ) = c1 c2 + c4 c5 − 2c3 ,  ∂t c4 − div(d4 (x)∇c4 ) = −c4 c5 + c3 ,    ∂t c5 − div(d5 (x)∇c5 ) = −c4 c5 + c3 ,      ∇ci · ν = 0, i = 1, 2, . . . , 5,    ci (x, 0) = ci,0 (x),

in Ω × R+ , in Ω × R+ , in Ω × R+ , in Ω × R+ , in Ω × R+ , on ∂Ω × R+ , in Ω.

23

(4.2)

where 0 < dmin ≤ di (x) ≤ dmax < +∞ for all x ∈ Ω and i = 1, 2, . . . , 5, are positive diffusion coefficients. The rest of this section is organized as follows: We first derive the mass conservation laws for (4.2), which play an essential role in our strategy. Then, we show that (4.2) satisfies the assumptions (A1) and (A2). Finally, we apply the strategy in Section 2 to show the explicit convergence to equilibrium for (4.2). For the sake of convenience, we will denote by c = (c1 , c2 , c3 , c4 , c5 ). We begin by stating the mass conservation laws whoses proof is straightforward and thus omitted. R Lemma 4.1 (Conservation laws). For i ∈ {1, 2} and j ∈ {4, 5}, we have (recall ci (t) = Ω ci (x, t)dx) ci (t) + cj (t) + c3 (t) = ci,0 + cj,0 + c3,0 =: Mi,j ,

for all

t > 0.

Among these four conservation laws, there are exactly three linear independent conservation laws. Moreover, in terms of the notations (2.2) and (2.8), the matrix Q can be chosen as   1 0 1 1 0     3×5  Q = 1 0 1 0 1 ∈R .   0 1 1 1 0 Remark 4.1. We denote by M = (M1,4 , M1,5 , M2,4 , M2,5 ) ∈ R4+ the vector of initial mass. Note that M is fixed once three linear independent of its four coordinates are prescribed. Lemma 4.2 (Detailed balance equilibrium). For any given positive initial mass M ∈ R4+ , there exists a unique positive equilibrium c∞ = (c1,∞ , c2,∞ , . . . , c5,∞ ) to (4.2) satisfying   c1,∞ c2,∞ = c3,∞ , c4,∞ c5,∞ = c3,∞ ,   ci,∞ + cj,∞ + c3,∞ = Mi,j , ∀i ∈ {1, 2}, ∀j ∈ {4, 5}. Consequently, the system (4.2) satisfies the assumptions (A1) and (A2). Proof. The proof of this Lemma follows similar calculations as the proof of Lemma 3.2 and is omitted.

To prove the convergence to the equilibrium for (4.2), we consider the entropy functional 5 Z X E(c) = (ci log ci − ci + 1)dx i=1

Ω

and its entropy-dissipation Z 5 Z X |∇ci |2 c1 c2 c4 c5 D(c) = di (x) dx + (c1 c2 − c3 ) log + (c4 c5 − c3 ) log dx. ci a3 c3 Ω i=1 Ω Proof of Theorem 1.2. We follow four steps in the strategy in Section 2. As mentioned already, it remains to prove Step 4. However, we would like to recall the main steps. Step 1. With E(c) − E(c∞ ) = (E(c) − E(c)) + (E(c) − E(c∞ )) and the Logarithmic Sobolev Inequality to estimate D(c) ≥ CLSI (dmin )(E(c) − E(c)), it remains to find a constant K1 > 0 such that 1 D(c) ≥ K1 (E(c) − E(c∞ )). 2

(4.3)

24


Step 2. By using the square root abbreviation, we have 5 X 1 D(c) ≥ 2dmin k∇Ci k2 + 2kC1 C2 − C3 k2 + 2kC4 C5 − C3 k2 2 i=1 2 5 q X E(c) − E(c∞ ) ≤ K2 Ci2 − Ci,∞ .

(4.4)

(4.5)

i=1

From (4.4) and (4.5), we get (4.3) provided that 2dmin

5 X

2

2

2

k∇Ci k + 2kC1 C2 − C3 k + 2kC4 C5 − C3 k ≥ K1 K2

i=1

5 q X

Ci2

2 − Ci,∞

.

(4.6)

i=1

Step 3. By applying Lemma 2.6, (4.6) holds provided 5 X

2 5 q K2 X 2 k∇Ci k + (C 1 C 2 − C 3 ) + (C 4 C 5 − C 3 ) ≥ K1 Ci − Ci,∞ K3 i=1 i=1 2

2

2

(4.7)

holds for a suitable constant K1 > 0. Step 4. We consider the ansatz 2 (1 + µi )2 , Ci2 = Ci,∞

µi ∈ [−1, +∞)

δi (x) = Ci (x) − Ci .

and

(4.8)

and recall that Ci = Ci,∞ (1 + µi ) − R(Ci )kδi k2

with

1 R(Ci ) = p 2 . Ci + Ci

For the following calculation, we shall choose ε > 0 such that ( ) 2 M1,4 M1,5 M2,5 M1,5 M1,4 M1,5 M2,5 2 ; ; ; . ε ≤ min ; ; 4 4 4 32M2,4 256M2,4 256

(4.9)

Case 1. Consider Ci2 ≥ ε2 for i = 1, 2, . . . , 5. Following the general approach in Subsection 2.2, we first choose θ ∈ (0, 1) such that 2 LHS of (4.7) ≥ θC3,∞ ((1 + µ1 )(1 + µ2 ) − (1 + µ3 ))2 + ((1 + µ4 )(1 + µ5 ) − (1 + µ3 ))2 and estimate RHS of (4.7) ≤ K1

5 X K2 2 max {Ci,∞ } µ2i . K3 i=1,...,5 i=1

Therefore, thanks to Lemma 4.3 we obtain (4.7) provided K1 ≤

2 θK3 C3,∞ 2 }. 12K2 maxi=1,...,5 {Ci,∞

Case 2. Consider Ci20 ≤ ε2 for some i0 ∈ {1, 2 . . . , 5}. We will first bound the right hand side of (4.7) above as 2 5 q 10K2 K K2 X Ci2 − Ci,∞ ≤ K1 . K1 K3 i=1 K3

(4.10)

(4.11)

Then, it is sufficient to show that the left hand side of (4.7) is bounded below by a positive constant. To do that, we will encounter two subcases due to various contributions of diffusion and reaction terms. (i) (Diffusion is dominant.) If kδi∗ k2 ≥ ε2 /2 for some i∗ ∈ {1, 2, . . . , 5}, then we can estimate LHS of (4.7) ≥ CP

5 X

kδi k2 ≥

i=1

CP ε2 . 2

Hence, (4.7) follows from (4.11) if we choose K1 ≤

K3 CP ε2 . 20KK2

(4.12)


25

(ii) (Diffusion is insufficient.) Consider kδi k2 ≤ ε2 /2 for all i = 1, 2, . . . , 5. We recall Ci20 ≤ ε2 for some i0 ∈ {1, 2, . . . , 5} and remark that the roles of C1 , C2 , C4 and C5 in (4.7) are the same. Therefore, withoug loss of generality, it is sufficient to investigate the two cases: i0 = 1 and i0 = 3:

When i0 = 1 then from the mass conservation C12 + C42 + C32 = M1,4

C12 + C52 + C32 = M1,5 ,

and

we get M1,4 M1,5 and C32 + C52 ≥ M1,5 − ε2 ≥ . (4.13) 2 2 Without loss of generality, we assume that M1,4 ≥ M1,5 . From (4.13) we have the following three possibilities C32 + C42 ≥ M1,4 − ε2 ≥

C32

Case

C42

C52

(I)

C32 ≥

M1,4 4

≤

M1,4 4

≤

M1,5 4

(II)

C32 ≤

M1,5 4

≥

M1,4 4

≥

M1,5 4

≥

M1,4 4

≤

M1,5 4

(III)

M1,5 4

≤ C32 ≤

In cases (I) and (III), we both have C32 ≥

M1,4 4

M1,5 4

and, thus

M1,5 ε2 M1,5 − ≥ . 4 2 8 Hence the left hand side of (4.7) is estimated as 2

C 3 = C32 − kδ3 k2 ≥

LHS of (4.7) ≥ (C 1 C 2 − C 3 )2 ≥

1 2 M1,5 M1,5 2 2 C − C 1C 2 ≥ − ε2 M2,4 ≥ 2 3 16 32

(4.14)

thanks to (4.9). In case (II), we have 2

C 4 = C42 − kδ4 k2 ≥

M1,4 8

and

2

C 5 = C52 − kδ5 k2 ≥

M1,5 . 8

We continue with 1 (C 4 C 5 − C 1 C 2 )2 2 1 2 2 1 2 2 M1,4 M1,5 1 M1,4 M1,5 ≥ C 4C 5 − C 1C 2 ≥ − ε2 M2,4 ≥ 4 2 256 2 512 thanks again to (4.9). Combining (4.14) and (4.15), we have M1,5 M1,4 M1,5 LHS of (4.7) ≥ min ; 32 512 LHS of (4.7) ≥ (C 1 C 2 − C 3 )2 + (C 4 C 5 − C 3 )2 ≥

(4.15)

(4.16)

which ends the proof in the case i0 = 1. 2 When i0 = 3, we obtain first that C 3 ≤ C32 ≤ ε2 . Without loss of generality, we can assume that M1,4 is the biggest component of M. Thus, C12 = C22 + M1,4 − M2,4 ≥ C22

and

C42 = C52 + M1,4 − M1,5 ≥ C52 .

By using the mass conservation C22 + C32 + C52 = M2,5 , we get C22 + C52 ≥ M2,5 − ε2 ≥ hence

M2,5 , 2

M2,5 M2,5 or C52 ≥ . 4 4 M then C12 ≥ 42,5 . It follows that C22 ≥

If C22 ≥

M2,5 4 2

C 1 = C12 − kδ1 k2 ≥

M2,5 8

and

2

C 2 = C22 − kδ2 k2 ≥

M2,5 . 8

26


We can then estimate due to (4.9) LHS of (4.7) ≥ (C 1 C 2 − C 3 )2 ≥ Similarly, if C52 ≥

M2,5 8 ,

2 2 M2,5 M2,5 1 2 2 2 C 1C 2 − C 3 ≥ − ε2 ≥ . 2 128 256

(4.17)

we can prove by using the same arguments above that LHS of (4.7) ≥

2 M2,5 , 256

(4.18)

which ends the case i0 = 3. Altogether from (4.16), (4.17) and (4.18), we get in Case 2. (ii), i.e. kδi k2 ≤ ε2 /2 for all i = 1, 2, . . . , 5 that ) ( 2 M1,5 M1,4 M1,5 M2,5 ; ; . (4.19) LHS of (4.7) ≥ min 32 512 256 Thus, from (4.19) and (4.11), we obtain (4.7) by choosing ) ( 2 K3 M1,5 M1,4 M1,5 M2,5 . min ; ; K1 ≤ 10KK2 32 512 256

(4.20)

Finally, the desired inequality (4.7) follows from (4.10), (4.12) and (4.20), which yields consequently the entropy entropy-dissipation estimate 1 D(c) ≥ λ(E(c) − E(c∞ )) with λ = min{CLSI (dmin ); 2K1 }. 2 Thus, by applying this functional inequality to solutions of (4.2), a Gronwall argument and the CiszárKullback-Pinsker inequality imply the following exponential convergence to equilibrium of solution: 5 X

−1 kci (t) − ci,∞ k2L1 (Ω) ≤ CCKP (E(c0 ) − E(c∞ ))e−λt

for all t > 0.

i=1

Lemma 4.3. Let µ1 , . . . , µ5 be defined as in (4.8). Then, 5 2 2 1 X 2 µ . (1 + µ1 )(1 + µ2 ) − (1 + µ3 ) + (1 + µ4 )(1 + µ5 ) − (1 + µ3 ) ≥ 12 i=1 i

(4.21)

Proof. This inequality is similar to (3.21). However, as mentioned, due to the different structure of mass conservation laws, we need to use a different proof. By the triangle inequality, the left hand side of (4.21) is bounded below by 2 2 (1 + µ1 )(1 + µ2 ) − (1 + µ3 ) + (1 + µ4 )(1 + µ5 ) − (1 + µ3 ) 2 2 1 ≥ (1 + µ1 )(1 + µ2 ) − (1 + µ3 ) + (1 + µ4 )(1 + µ5 ) − (1 + µ3 ) 4 | {z } | {z } =:H1

=:H2

2 + (1 + µ1 )(1 + µ2 ) − (1 + µ4 )(1 + µ5 ) | {z }

(4.22)

=:H3

2 2 = C22 − C2,∞ , we have From the conservation law C12 − C1,∞ 2 2 C1,∞ µ1 (µ1 + 2) = C2,∞ µ2 (µ2 + 2).

Moreover, sonce µ1 , µ2 ∈ [−1, +∞), it follows that µ1 and µ2 have always a same sign. Similarly, µ4 and µ5 have always a same sign. We have the following useful estimates • If µ1 and µ3 have different signs, then 1 H1 ≥ (µ21 + µ22 + µ23 ). (H1) 2 Indeed, thanks to µ1 µ3 ≤ 0 ≤ µ1 µ2 and (1 + µ1 ) ≥ 0, we have (µ1 − µ3 )µ2 (1 + µ1 ) ≥ 0. Hence H1 = [µ1 − µ3 + µ2 (1 + µ1 )]2 ≥ (µ1 − µ3 )2 ≥ µ21 + µ23 . Similarly we get H1 ≥ µ22 + µ23 and consequently that the estimate (H1) holds.


27

• If µ4 and µ3 have different signs then H2 ≥

1 2 (µ + µ25 + µ23 ). 2 4

(H2)

This can be proved in the same way as (H1). • If µ1 and µ4 have different signs then H3 ≥

1 2 (µ + µ22 + µ24 + µ25 ). 3 1

(H3)

Note that in this case µ1 µ4 ≤ 0 ≤ µ1 µ2 , µ4 µ5 and (1 + µ1 ), (1 + µ4 ) ≥ 0 and we have always (µ1 − µ4 )µ2 (1 + µ1 ) ≥ 0,

−(µ1 − µ4 )µ5 (1 + µ4 ) ≥ 0

and

− µ2 µ5 (1 + µ1 )(1 + µ4 ) ≥ 0.

Therefore, H3 = [µ1 − µ4 + µ2 (1 + µ1 ) − µ5 (1 + µ4 )]2 ≥ (µ1 − µ4 )2 ≥ µ21 + µ24 . Similarly we have H3 ≥ µ21 + µ25 and H3 ≥ µ22 + µ24 and consequently that (H3) holds. 2 2 2 + C3,∞ From the mass conservation C12 + C32 + C42 = M1,4 = C1,∞ + C4,∞ , we obtain 2 2 2 C1,∞ µ1 (µ1 + 2) + C3,∞ µ3 (µ3 + 2) + C4,∞ µ4 (µ4 + 2) = 0,

which causes the following possibilities concerning the signs of µ1 , µ3 and µ4 . Case

µ1

µ3

µ4

(I)

-

-

+

(II)

-

+

-

(III)

-

+

+

(IV)

+

-

-

(V)

+

-

+

(VI)

+

+

-

We see that in all these cases, there are always two of the three estimates (H1), (H2) and (H3) that hold. Therefore, thanks to (4.22), we obtain the desired estimate (4.21). 5. Summary, further applications and open problems In this paper, we exploit the entropy method to show explicit convergence to equilibrium for detailed balance chemical reaction-diffusion networks describing substances in a bounded domain Ω ⊂ Rn according to the mass action law. More precisely, in Section 2.2 we propose a constructive method to prove an entropy entropy-dissipation estimate with computable rates consisting of four steps in which the first three steps are rigorously verified and the last step suggests a clear way to proceed for a specifically given system, where the mass conservation laws are explicitly known. This is demonstrated in Sections 3 and 4 for two specific types of reaction-diffusion networks: A single reversible reaction (3.1) and a chain of reversible reactions (4.2), which is motivated by enzyme reactions. 5.1. Further applications. We point out that the proposed approach applies also to reaction-diffusion networks where the chemical substances are supported on different spatial compartments. For example in [FLT], the reversible reaction αU βV is considered between a bounded domain Ω ⊂ Rn and its smooth boundary ∂Ω, where U is the volumesubstance inside Ω and V is the surface-substance on ∂Ω and the reaction is assumed to happen on ∂Ω.

28


The corresponding volume-surface reaction-diffusion system reads as  ut − du ∆u = 0, x ∈ Ω, t > 0,    d ∂ u = −α(uα − v β ), x ∈ ∂Ω, t > 0, u ν α β  vt − dv ∆∂Ω v = β(u − v ), x ∈ ∂Ω, t > 0,    u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω,

(5.1)

in which u : Ω × R+ → R+ is the volume-concentration of U and v : ∂Ω × R+ → R+ is the surfaceconcentration of V, and ∆∂Ω is the Laplace-Beltrami operator which describes diffusion of V along ∂Ω. The system (5.1) possesses the mass conservation law Z Z Z Z β u(x, t)dx + α v(x, t)dS = β u0 (x)dx + α v0 (x)dS =: M > 0, ∀t≥0 Ω

∂Ω

Ω

Γ

and thus obeys a unique positive equilibrium (u∞ , v∞ ) satisfying ( β uα ∞ = v∞ , β|Ω|u∞ + α|Γ|v∞ = M. To show the convergence to equilibrium for (5.1), we consider the entropy functional Z Z E(u, v) = (u log u − u + 1)dx + (v log v − v + 1)dS Ω

Γ

and its entropy-dissipation Z Z uα |∇Γ v|2 |∇u|2 dx + du dS + (uα − v β ) log β dS. D(u, v) = du u v v Γ Γ Ω The aim is to prove an EED estimate of the form Z

D(u, v) ≥ λ(E(u, v) − E(u∞ , v∞ )), R R for all (u, v) satisfying the mass conservation β Ω u(x)dx + α Γ v(x)dS = M .

(5.2)

The EED estimate (5.2) can be proved by applying the method proposed in Section 2 with few changes, e.g. the Poincaré inequality is replaced by the Trace inequality k∇f k2L2 (Ω) ≥ CT kf − f k2L2 (∂Ω) . The interested reader is referred to [FLT] for more details. In fact, it’s easy to verify that the results of [FLT] are also applicable in the following network α1 U1 + . . . + αI UI β1 V1 + . . . + βJ VJ in the case that Ui=1,...,I are volume-concentrations on Ω and Vj=1,...,J are surface-concentrations on ∂Ω. 5.2. Open problems. There are potentially various open problems relating to the systems considered in this paper. Among them, we list here the two questions we find the most interesting: 1. How to choose the conservation laws in the general case? As mentioned in the introduction, the conservation laws Q c = M depend on the choice of the matrix Q, which has rows forming a basis of ker(W ), where W is the Wegscheider matrix. The choice of Q is not unique and in fact, there are infinitely many matrices like Q. The question is: can we have a procedure or a method to choose such a matrix Q for general systems, which is suitable for our method and allows to complete the proof of Step 4 in the general case? 2. How to get optimal convergence rate? We made it clear in this paper (see Remark 2.3) that although we obtain an explicit bound for the convergence rate, the convergence rate in this work is non-optimal. The question of optimal convergence rate using the entropy method is more involved and left for future investigation. Acknowledgements. The second author is supported by International Research Training Group IGDK 1754. This work has partially been supported by NAWI Graz. References [AMTU01] A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for FokkerPlanck type equations, Comm. Partial Differential Equations 26 (2001) 43–100. [BCD07] M. Bisi, F. Conforto, L. Desvillettes, Quasi-steady-state approximation for reaction–diffusion equations, Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007) 823–850. [BP10] D. Bothe, M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl. 368 (2010) 120–132.


29

[CDF08] J.A. Carrillo, L. Desvillettes, K. Fellner, Exponential decay towards equilibrium for the inhomogeneous AizenmanBak model, Commun. Math. Phys., 278 (2008), 433–451. [CDF08a] J.A. Carrillo, L. Desvillettes and K. Fellner, Fast-Reaction Limit for the Inhomogeneous Aizenman-Bak Model, Kinet. Relat. Models 1 (2008) 127–137. [CF06] G. Craciun, M. Feinberg, Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph, SIAM J. Appl. Math. 66 (2006) 1321–1338. [CV03] J. Carrillo, J.L. Vazquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), 1023–1056. [CDSS09] G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551–1565. [Cra] G. Craciun, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture, arXiv:1501.02860. [PD02] M. Del Pino, J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. 81 (2002), 847–875. [DF06] L. Desvillettes, K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl. 319 (2006) 157–176. [DF08] L. Desvillettes, K. Fellner, Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds, Revista Matematica Iberoamericana 24 (2008) 407–431. [DFM08] M. Di Francesco, K. Fellner, P. Markowich, The entropy dissipation method for inhomogeneous reaction–diffusion systems, Proc. Royal Soc. A 464 (2008) 3272–3300. [DF14] L. Desvillettes, K. Fellner, Exponential Convergence to Equilibrium for a Nonlinear Reaction-Diffusion Systems Arising in Reversible Chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014) 96–104. [DFT] L. Desvillettes, K. Fellner, B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, Preprint 2016. [DV00] L. Desvillettes, C. Villani, On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications, Comm. Partial Differential Equations 25, no. 1-2 (2000), 261–298. [DV01] L. Desvillettes, C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54, 1 (2001), 1–42. [DV05] L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones Mathematicae 159, (2005) 245–316. [FL16] K. Fellner, E.-H. Laamri, Exponential decay towards equilibrium and global classical solutions for nonlinear reactiondiffusion systems, to appear in J. Evol. Equ. (2016). [FLT] K. Fellner, E. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volume-surface reactiondiffusion system with nonlinear boundary coupling, arXiv:1404.2809. [FNS04] K. Fellner, L. Neumann, C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatsh. Math. 141 (2004), 289–299. [Fis15] J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015) 553–587. [GZ10] I. Gentil, B. Zegarlinski, Asymptotic behaviour of a general reversible chemical reaction-diffusion equation, Kinet. Relat. Models 3 (2010), 427–444. [GGH96] A. Glitzky, K. Gr¨ oger, R. H¨ unlich, Free energy and dissipation rate for reaction-diffusion processes of electrically charged species, Appl. Anal. 60 (1996), 201–217. [GH97] A. Glitzky, R. H¨ unlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems, Z. Angew. Math. Mech. 77 (1997), 823–832. [Pie10] Pierre, M., Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78 (2010), 417–455. [MHM15] A. Mielke, J. Haskovec, P. A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dyn. Diff. Eqs. 27 (2015) 897–928. [Mur02] James D. Murray, Mathematical Biology, Springer, 2002. [TV99] G. Toscani, C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Commun. Math. Phys. 203 (1999), 667–706. [TV00] G. Toscani, C. Villani, On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds, J. Statist. Phys. 98 (2000), 1279–1309. [Vil03] C. Villani, Cercignani’s conjecture is sometimes true and always almost true, Commun. Math. Phys. 234 (2003), 455–490. [VVV94] A.I. Volpert, V.A. Volpert, VL.A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, RI, 1994. Bao Quoc Tang1,2 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria 2 Faculty of Applied Mathematics and Informatics, Hanoi University of Sience and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam E-mail address: [email protected] 1

Klemens Fellner Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraβe 36, 8010 Graz, Austria E-mail address: [email protected]

EXPLICIT EXPONENTIAL CONVERGENCE TO ... - Uni Graz

EXPLICIT EXPONENTIAL CONVERGENCE TO ... - Uni Graz

Suggest Documents

Uni Graz Tipps - Karl-Franzens-Universität Graz

Inventorying PM emissions - Uni Graz

Evaluation of a dynamic-diagnostic modelling approach to ... - Uni Graz

EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL

9. Exponential maps and explicit formulas - arXiv

Welcome to Graz - Karl-Franzens-Universität Graz

EXPONENTIAL CONVERGENCE OF A LINEAR RATIONAL ...

EXPONENTIAL CONVERGENCE OF SOLUTIONS OF SICNNS WITH

Exponential convergence rate in Boltzmann-Shannon entropy

Exponential Convergence of Multiquadric Collocation Method: a

Exponential Convergence to Equilibrium for a ... - Semantic Scholar

Convergence of MLE to MVUE of reliability for exponential ... - ijritcc

General Robust Bayes Pseudo-Posterior: Exponential Convergence ...

Computable Exponential Convergence Rates for Stochastically ...

Exponential rate of convergence in current reservoirs

Total Variation Regularization for Nonlinear Fluorescence ... - Uni Graz

Relabelling the preserved speech variant of Rett ... - Med Uni Graz

Interaction of robot swarms using the honeybee-inspired ... - Uni Graz

InputâOutput Analysis from a Wider Perspective - Uni Graz

Early Neurological Signs in Preterm Infants with ... - Med Uni Graz

General Movements Detect Early Signs of Hemiplegia ... - Med Uni Graz

Explicit To

Why to Worry about the Wealth of Nations - online .uni-graz .at

Explicit Exponential Rosenbrock Methods and their Application in ...

EXPLICIT EXPONENTIAL CONVERGENCE TO ... - Uni Graz