Explicit exponential convergence to equilibrium for nonlinear reaction-diffusion systems with detailed balance condition

0.45 MB·

by  Klemens Fellner

EXPLICIT EXPONENTIAL CONVERGENCE TO EQUILIBRIUM
FOR NONLINEAR REACTION-DIFFUSION SYSTEMS
WITH DETAILED BALANCE CONDITION
KLEMENSFELLNER,BAOQUOCTANG
Abstract. Theconvergencetoequilibriumofmassactionreaction-diffusionsystemsarisingfromnet-
works of chemical reactions is studied. The considered reaction networks are assumed to satisfy the
detailed balance conditionand haveno boundaryequilibria. Wepropose ageneral approach basedon
6 the so-called entropy method, which is able to quantify with explicitly computable rates the decay of
1 anentropyfunctionalintermsofanentropyentropy-dissipationinequalitybasedonthetotalityofthe
0 conservationlawsofthesystem.
2 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
g
systems: asinglereversiblereactioninvolvinganarbitrarynumberofchemicalsubstances andachain
u
oftworeversiblereactionsarisingfromenzymereactions.
A

7
2
Contents
]
P
1. Introduction and main results 1
A
2. The general method 9
h. 2.1. Preliminary estimates and inequalities 10
t 2.2. A constructive method to prove the EED estimate 12
a
m 3. A single reversible reaction - Proof of Theorem 1.2 18
4. Reversible enzyme reactions - Proof of Theorem 1.3 22
[
5. Summary, further applications and open problems 26

2 5.1. Further applications 26
v
5.2. Open problems 27
2
References 27
9
9
5
0
. 1. Introduction and main results
1
0
In this paper, we study exponential convergence to equilibrium with explicitly bounded rates and
6
constants for reaction-diffusion systems arising from chemical reaction networks.
1
: The considered reaction-diffusion systems describe networks of chemical reactions according to the
v
i law of mass action kinetics and under the assumption of the detailed balance condition. More precisely,
X
we consider I chemical substances ,..., reacting in R reversible reactions of the form
1 I
C C
r
a αr +...+αr kbr βr +...+βr
1C1 ICI kr 1C1 ICI
f
for r = 1,2,...,R with the nonnegative stoichiometric coefficients αr = (αr,...,αr) ( 0 [1, ))I
and βr =(βr,...,βr) ( 0 [1, ))I and the positive forward and backw1ard reaIctio∈n r{at}e∪cons∞tants
1 I ∈ { }∪ ∞
kr > 0 and kr > 0. The corresponding reaction-diffusion system for the concentration vector c =
f b
(c ,...,c ):Ω [0,+ ) [0,+ )I subject to homogeneous Neumann boundary conditions reads as
1 I
× ∞ → ∞
∂
c=D∆c R(c), in Ω R ,
+
∂t − ×
c ν =0, on ∂Ω R , (1.1)
+
∇ · ×
c(x,0)=c (x), on Ω,
0
2010 Mathematics Subject Classification. 35B35,35B40,35K57, 35Q92.
Keywordsandphrases. Reaction-DiffusionSystems;ExponentialConvergencetoEquilibrium;EntropyMethod;Chem-
icalReactionNetworks;DetailedBalanceCondition.
1
2 K.FELLNER,B.Q.TANG
where Ω Rn is abounded domainwith smoothboundary∂Ω (e.g. ∂Ω C2+ǫ,ǫ>0),outwardnormal
⊂ ∈
unit vector ν and normalised volume, i.e.
Ω =1
| |
(this isw.l.o.g. byrescalingthe positionvariablex Ωasx xΩ1/n). Moreover,D=diag(d ,...,d )
1 I
∈ → | |
is a uniformly positive definite diffusion matrix, i.e. 0 < d d d < + for all i = 1,...,I,
min i max
≤ ≤ ∞
and the reaction vector R(c) represents the chemical reactions according to the mass action law, i.e.
R I
R(c)= (αr βr) krcαr krcβr with cαr = cαri for r =1,2,...,R. (1.2)
− f − b i
Xr=1 (cid:16) (cid:17) iY=1
Here, by following e.g. [VVV94], we observe that the reaction vector R(c) can be written as product of
the stoichiometric matrix
W =((βr αr)r=1,...,R)⊤ RR×I,
− ∈
which is also called Wegscheider matrix, and the reaction rates vector K(c) as modelled according to
the mass action law, i.e.
R(c)=−W⊤K(c), where K(c)=(Kr(c))r=1,...,R := kfrcαr −kbrcβr r=1,...,R.
(cid:16) (cid:17)
The range rg(W ) is called the stoichiometric subspace and the above implies that R(c) rg(W ).
⊤ ⊤
∈
As a consequence, a key structural property of the reaction vector R(c) is the codim of W, which we
denote by m=dimker(W).
If m > 0, then there exists a (non-unique) matrix Q Rm I of zero left-eigenvectors such that
×
∈
QR(c) = 0 for all states c. As a consequence, we have the following mass conservation laws for (1.1)–
(1.2)
Qc(t)dx= Qc dx or equivalently Qc(t)=M:=Qc for all t>0, (1.3)
0 0
ZΩ ZΩ
where c = (c ,...,c ) with c (t) = c (x,t)dx is the spatially averaged concentration vector (recall
1 I i Ω i
Ω =1) and M denotes the vector of initial masses, which can be assumed non-negative, i.e. M Rm
(|af|ter changing the sign of the rowsRof Q, for which the corresponding component of M shou∈ld ≥be0
negative). If m=0, then the system (1.1)–(1.2) has no conservation law.
Moreover, it is well known that the reaction vector R(c) according to the mass action law, satisfies
the quasi-positivity condition: If R(c)=(R (c),...,R (c)) , then
1 I ⊤
i=1,2,...I : R (c ,...,c ,0,c ,...,c ) 0 for all c ,...,c ,c ,...,c 0.
i 1 i 1 i+1 I 1 i 1 i+1 I
∀ − ≥ − ≥
Asaconsequence,solutionsto(1.1)-(1.2)subjecttonon-negativeinitialdatac 0remainnon-negative
0
≥
c(t) 0 for all times t>0, see e.g. [Pie10].
≥
The first main assumption concerning that reactions networks considered in this work is the detailed
balance condition, see e.g. [HJ72, Vol72, Fei79, VVV94]: A state c [0,+ )I is called a homogeneous
∗
∈ ∞
equilibrium or shortly an equilibrium of (1.1)–(1.2) if and only if
R(c )=0 and Qc =M.
∗ ∗
(A1) System (1.1)–(1.2) is assumed to satisfy the detailed balance condition, that is there exists an
equilibrium c (0,+ )I such that any forward reaction is balanced with its corresponding
∞ ∈ ∞
backwardreaction at this equilibrium, i.e.
krcαr =krcβr, for all r =1,2,...,R.
f b
∞ ∞
This equilibrium c is called a detailed balance equilibrium.
∞
Thedetailedbalanceconditionallowstorescalethe system(1.1)–(1.2)suchthatwemayassumew.l.o.g.
kr =kr =kr >0, for all r =1,2,...,R. (1.4)
f b
Thisrescalingsimplifiestheformulationofasecondcrucialconsequenceofthedetailedbalancecondition
that the logarithmic entropy (or free energy) functional, which is the key quantity of our study,
I
(c)= (c logc c +1)dx, (1.5)
i i i
E −
i=1ZΩ
X
decays monotone in time according to the following entropy-dissipationfunctional
d I c 2 R
(c)= (c)= d |∇ i| dx+ kr (cαr cβr)(logcαr logcβr)dx 0. (1.6)
i
D −dtE c − − ≥
i=1ZΩ i r=1 ZΩ
X X
CONVERGENCE TO EQUILIBRIUM FOR REACTION-DIFFUSION SYSTEMS 3
It is well known for detailed balanced reaction networks (see e.g. [HJ72] for the ODE systems and
[GGH96, Lemma 3.4] for reaction-diffusion systems with homogeneous Neumann boundary conditions)
that for a given positive initial mass vector M Rm, there exists a unique positive detailed balance
∈ +
equilibrium c = (c ,...,c ) of (1.1), which is the unique vector of positive constants c > 0
1, I,
∞ ∞ ∞ ∞
balancing all the reactions and satisfying the mass conservation laws, i.e.
c >0 : cαr =cβr for all r =1,2,...,R and Qc =M. (1.7)
∞ ∞ ∞ ∞
Note that the existence of a positive detailed balance equilibrium c >0 cantypically also be expected
for non-negative initial mass vectors M Rm with exceptions whe∞n M =0. However, the (chemically
meaningless) reversible reaction 2∈ +≥0 constitutes an example with positive detailed balance
1 1 2
C ↔ C C
equilibrium c =1 in the case when the initial mass vector M=0 since Q=(1, 1) for this system.
∞ −
It is important to remark that besides the unique positive detailed balance equilibrium c > 0,
∞
there may also exist additional, so-called boundary equilibria, for which c = 0 for at least one index
∗i,
i=1,...,I. See Remark 2.1 for an example of a system having a boundary∞equilibrium.
In this paper, we only consider systems with positive detailed balance equilibrium (1.7) and without
boundary equilibria. We therefore impose the following second equilibrium assumption:
(A2) The system (1.1)–(1.2) features no boundary equilibrium, that is (1.1)–(1.2) does not possess an
equilibrium c ∂[0,+ )I.
∗
∈ ∞
Assumption (A2) is a natural structural condition in order to prove an entropy entropy-dissipation
estimate like presented in the following and stated in (1.9). In fact, for general systems featuring
boundary equilibria, the behaviour near a boundary equilibrium is unclear and boundary equilibria are
abletopreventglobalexponentialdecaytoanasymptoticallystablepositivedetailedbalanceequilibrium,
see e.g. [DFT]. It is also remarked that there exists a large class of systems possessing no boundary
equilibria. See e.g. [CF06] for necessary conditions to determine such systems.
Thelargetimebehaviourofsolutionstononlinearreaction-diffusionsystemsisahighlyactiveresearch
area,whichposes many openproblems. Classicalmethods include e.g. linearisationtechniques,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 (f) and its nonnegative entropy-dissipation functional
E
d
(f)= (f(t)) 0
D −dtE ≥
along the flow of a PDE model, which is well-behaved in the following sense: firstly, all states satisfying
(f)=0 as well as all the involved conservation laws identify a unique entropy-minimising equilibrium
D
f , i.e.
∞
(f)=0 and all conservation laws f =f ,
D ⇐⇒ ∞
and secondly, there exists an entropy entropy-dissipation (EED for short) estimate of the form
(f) Φ( (f) (f )), with Φ(x) 0, Φ(x)=0 x=0,
D ≥ E −E ∞ ≥ ⇐⇒
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 Φ(0) = 0, a Gronwall argument usually
′
6
implies exponential convergence toward f in relative entropy (f) (f ) with a rate, which can
∞ E −E ∞
be explicitly estimated. Finally, by applying Csisza´r-Kullback-Pinsker type inequalities to the relative
entropy (f) (f ) (recall that (f) is convex), one obtains exponential convergence to equilibrium,
for instanEce, w−.rE.t. ∞the L1-norm. E
The entropy method is a fully nonlinear alternative to arguments based on linearisation around the
equilibriumandhastheadvantageofbeingquiterobustwithrespecttovariationsandgeneralisationsof
themodelsystem. Thisisduetothefactthattheentropymethodreliesmainlyonfunctionalinequalities
whichhavenodirectlinktotheoriginalPDEmodel. Generalisedmodelstypicallyfeaturerelatedentropy
and entropy-dissipation functionals. Thus, previously established EED estimates may very usefully be
re-applied.
Theentropymethodhaspreviouslybeenusedforscalarequations: nonlineardiffusionequations(such
as fast diffusions [CV03, PD02], Landau equation [DV00]), integral equations (such as the spatially
4 K.FELLNER,B.Q.TANG
homogeneous Boltzmann equation [TV99, TV00, Vil03]), kinetic equations (see e.g. [DV01, DV05,
FNS04]),orcoagulation-fragmentationmodels(seee.g. [CDF08,CDF08a]). Forcertainsystemsofdrift-
diffusion-reaction equations in semiconductor physics, an entropy entropy-dissipationestimate has been
shownintwodimensionsindirectlyviaacompactness-basedcontradictionargumentin[GGH96,GH97].
The first results for EED estimates for reaction-diffusion systems with explicit rates and constants
were established in [DF06, DF08, DFM08, GZ10, DF14] in particular cases of reversible equations with
(at most) quadratic nonlinearities.
In this paper, we aim to generalise the entropy method to reaction-diffusion systems with arbitrary
mass action law nonlinearities and, as a consequence, show exponential convergence to equilibrium for
(1.1) with explicit bounds on the rates and constants. The obtained results extend recent works on the
convergence to equilibrium for nonlinear chemical reaction-diffusion systems, see e.g. [MHM15, FL16,
FLT].
Beforestatingourresults,weremarkthatforgeneralnonlinearreaction-diffusionsystemsofthe form
(1.1)–(1.2), the existence of global classical or weak solutions is often an open problem, especially in
higher space dimensions and for super-quadratic nonlinearities (see e.g. the survey [Pie10] and Remark
1.1 for a more detailed discussion). This is due to the lack of sufficiently strong a-priori estimates in
order to control nonlinear terms (comparison principles do not hold except for special systems).
Recently, Fischer [Fis15] proved the global existence of so-called ”renormalised solution” for general
reaction-diffusion systems, which dissipate the entropy (1.5), and thus provided the existence of global
renormalised solutions of system (1.1)–(1.2) for arbitrary stoichiometric coefficients.
Proposition 1.1 (Global renormalised solutions to mass action reaction-diffusion systems [Fis15]).
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Assume that the diffusion matrix D is
positive definite, i.e. d > 0 for all i = 1,...,I. Let c L1(Ω)I be nonnegative initial data with
i 0
∈
(c )<+ .
0
E ∞
Then, there exists a global in time non-negative renormalised solution c to (1.1)–(1.2), i.e. 0 c
i
≤ ∈
L∞loc([0,+∞);L1(Ω)) and √ci ∈ L2loc([0,+∞);H1(Ω)), and for every smooth function ξ : (R+)I → R
with compactly supported derivative ξ and for every testfunction ψ C (Ω [0,+ )) the equation
∞
∇ ∈ × ∞
T d
ξ(c(,T))ψ(,T)dx ξ(c )ψ(,0)dx ξ(c) ψdxdt
0
· · − · − dt
ZΩ ZΩ Z0 ZΩ
I T
= ψ∂ ∂ ξ(c)(d c ) c dxdt
i j i i j (1.8)
− ∇ ·∇
i,j=1Z0 ZΩ
X
I T I T
∂ ξ(c)(d c ) ψdxdt+ ∂ ξ(c)R (c)ψdxdt
i i i i i
− ∇ ·∇
i=1Z0 ZΩ i=1Z0 ZΩ
X X
holds for almost every T >0, with R(c)=(R (c),...,R (c)).
1 I
Note that the regularity of renormalised solutions is in general insufficient to guarantee the L1-
integrability of the reaction-terms on the right hand side of (1.1) and in the entropy dissipation (1.6).
However, sufficiently integrable renormalisedsolutions are weak solutions, see e.g. [DFPV07, Fis15].
Inthepresentpaper,weshowresultsoftheformthatifarenormalisedsolutionisassumedtosatisfies
a weak entropy entropy-dissipation law (see (1.10) below), then the entropy method implies exponential
convergence of the renormalised solution to equilibrium. This includes the cases when renormalised
solutions are in fact classical or weak solutions, for which the weak entropy entropy-dissipation law
(1.10) can be shown to hold rigorously.
The key lemma of the entropy method is the following EED functional inequality
(c) λ( (c) (c )) (1.9)
D ≥ E −E ∞
for all c L1(Ω;[0,+ )I) obeying the mass conservation Qc = M, and where λ = λ(Ω,D,M,αr,βr)
∈ ∞
is a constantdepending onthe domainΩ, the diffusion coefficients D, the initial mass vector M andthe
stoichiometric coefficients.
Let’s assume for the moment that the functional inequality (1.9) is proven. Assume moreover the
existence of (classical, weak or renormalised) solutions of the reaction-diffusion system (1.1), which are
supposed to satisfy the following weak entropy entropy-dissipation law as a generalised version of (1.6)
t
(c(t))+ (c(s))ds (c ) for a.a. t>0. (1.10)
0
E D ≤E
Z0
CONVERGENCE TO EQUILIBRIUM FOR REACTION-DIFFUSION SYSTEMS 5
Then, by applying the functional inequality (1.9) to such solutions of (1.1)–(1.2), a Gronwall argument
(see e.g. [Wil, FL16]) yields exponential convergence in relative entropy with the rate λ, where λ is
given in (1.9) and can be explicitly estimated. Moreover, by applying a Csisza´r-Kullback-Pinsker type
inequality (see Section 2) one also obtains L1-convergence to equilibrium of solutions to (1.1) with the
rate e λt/2.
−
In[MHM15], by usinganinspiredconvexificationargumentandunderthe assumptionofthe detailed
balance condition, the authors proved that (1.9) holds for system (1.1)–(1.2) for a λ > 0 provided that
the detailed balance equilibrium (1.7) is the only equilibrium and that there are no boundary equilibria.
Moreover, they gave an explicit bound of λ in the case of the quadratic reaction 2 ⇌ . However,
1 2
C C
becauseofthenon-convexstructureofthe problem,obtainingexplicitestimatesonλviaconvexification
seems difficult in the case of more than two substances, e.g. for systems like
α +β ⇌γ or + ⇌ + .
1 2 3 1 2 3 4
C C C C C C C
By drawing from various previous ideas in [DF08, DF14, FL16, FLT], this paper aims to propose a
constructive way to prove quantitatively the EED estimate (1.9) 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 advantage of the here-proposed method is its robustness in the sense that it also applies
to (bio-)chemical reaction networks where substances are supported on different compartments. In
Subsection 5.1, we present that our method directly generalises to a specific example of a volume-
surfacereaction-diffusionsystem. Volume-surfacereaction-diffusionsystemsarerecentlybecominghighly
relevantmodelswithmanyapplicationsincell-biology[NGCRSS07,MS11,FNR13,FRT16,FLT]oralso
crystal growth [KD01]. We remark that the method of convexification as presented in [MHM15] seems
not to apply to such volume-surface reaction-diffusion systems.
The first two main results of this paper will detail the proposed method for two important model
systems: a general single reversible reaction with arbitrary number of substances, i.e.
α +...+α ⇌β +...+β (1.11)
1 1 I I 1 1 J J
A A B B
andachainoftworeversiblereactions,whichgeneralisestheMichaels-Mentonmodelforcatalyticenzyme
kinetics (see e.g. [Mur02])
+ ⇌ ⇌ + . (1.12)
1 2 3 4 5
C C C C C
Note that for the single reversible reaction (1.11), it is more convenient and consistent with the
literature to change the notation compared to (1.1) by splitting the concentration vector c into a left-
hand-side and a right-hand-side-concentrationvector, i.e.
c=(c ,...,c ) (a,b)=(a ,...,a ,b ,...,b ),
1 I 1 I 1 J
→
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.
Atfirst,afterassuming(w.l.o.g.) thattheforwardandbackwardreactionrateconstantsarenormalised
to one, the mass action reaction-diffusion system modelling (1.11) reads as
∂ a d ∆a = α (aα bβ), in Ω R , i=1,2,...,I,
t i a,i i i +
− − − ×
∂ b d ∆b =β (aα bβ), in Ω R , j =1,2,...,J,
 t j − b,j j j − × + (1.13)
∇ai·ν =∇bj ·ν =0, on ∂Ω×R+, i=1,...,I, j =1,...,J,
a(x,0)=a (x), b(x,0)=b (x), in Ω,
0 0
where a=(a1,...,aI) and b=(b1,...,bJ) denote the two vectors for left- and right-hand side concen-
trations, d >0 and d >0 are the positive diffusion coefficients, and α=(α ,...,α ) [1, )I and
a,i b,j 1 I
∈ ∞
β = (β ,...,β ) [1, )I are the positive vectors of the stoichiometric coefficients assossiated to the
1 J
∈ ∞
single reaction (1.11). Recall that aα = I aαi and bβ = J bβj.
i=1 i j=1 j
The system (1.13) possesses the following IJ mass conservation laws
Q Q
a b
i j
+ =M , i=1,...,I, j =1,...,J, (1.14)
i,j
α β
i j
fromwhichexactlym=I+J 1conservationlawsarelinearindependent. ThatmeansthematrixQin
−
this case has the dimension Q R(I+J 1) (I+J). See Lemma 3.1 below for an explicit form of Q. After
− ×
∈
choosingandfixingI+J 1linearindependentcomponentsfromtheIJ conservedmasses(M ) RIJ,
− i,j ∈ +
6 K.FELLNER,B.Q.TANG
wedenotebyM RI+J 1 thevectorofinitialmassescorrespondingtotheselectedI+J 1coordinates
−
∈ −
of (M ) RIJ. Thus, by a giveninitial mass vector M, we signify that these I+J 1 coordinates are
i,j ∈ + −
given and the remaining coordinates of (M ) RIJ are subsequently calculated. The unique positive
i,j ∈ +
detailed balance equilibrium (a ,b ) RI+J of (1.13) is defined by
∞ ∞ ∈ +
ai,∞ + bj,∞ =M i=1,2,...,I, j =1,2,...,J,
αi βj i,j ∀ ∀
aα =bβ .

∞ ∞
The corresponding entropy and entropy-dissipationfunctionals for system (1.13) are

I J
(a,b)= (a loga a +1)dx+ (b logb b +1)dx (1.15)
i i i j j j
E − −
i=1ZΩ j=1ZΩ
X X
and
I a 2 J b 2 aα
(a,b)= d |∇ i| dx+ d |∇ j| dx+ (aα bβ)log dx, (1.16)
D i=1ZΩ a,i ai j=1ZΩ b,j bj ZΩ − bβ
X X
respectively.
Theorem 1.2 (Explicit convergence to equilibrium for a general single reversible reaction (1.11)).
Let Ω Rn be a bounded domain with smooth boundary. Assume positive diffusion coefficients d >0
a,i
⊂
and d >0 for all i=1,...,I and j =1,...,J. Assume stoichiometric coefficients α 1 and β 1
b,j i j
≥ ≥
for all i=1,...,I and j =1,...,J.
Supposeassumptions(A1)and(A2),i.e. thatsystem (1.13)featuresauniquepositivedetailedbalance
equilibrium (a ,b ) of the form (1.7) and no boundary equilibria. Note that for a given positive initial
mass vector M∞∈ ∞R+I+J−1 corresponding to I +J −1 linear independent conservation laws (1.14), the
assumptions (A1) and (A2) are in fact a consequence of Lemma 3.2 and thus satisfied.
Then, for any nonnegative (a,b) L1 Ω;[0, )I+J satisfying the mass conservation laws (1.14),
∈ ∞
the following EED estimate
(cid:0) (cid:1)
(a,b) λ ( (a,b) (a ,b )) (1.17)
1
D ≥ E −E ∞ ∞
holds for (a,b) and (a,b) defined in (1.15) and (1.16), respectively, and where the constant λ > 0
1
E D
can be explicitly estimated in terms of the initial mass vector M, the domain Ω, the stoichiometric
coefficients α,β and the diffusion coefficients d and d .
a,i b,j
Moreover, for any nonnegative initial data (a ,b ) L1(Ω)I+J with finite entropy, i.e. (a ,b ) <
0 0 0 0
∈ E
+ , there exists a global non-negative renormalised solution (a,b) of (1.1)–(1.4) in the sense of Propo-
∞
sition 1.1. Assume that this renormalised solution satisfies the weak entropy entropy-dissipation law
(1.10), i.e.
t
(a(t),b(t))+ (a(s),b(s))ds (a ,b ) for a.a. t>0.
0 0
E D ≤E
Z0
Then, thisrenormalisedsolutionof (1.1)–(1.4)convergesexponentiallytothedetailedbalanceequilibrium
(a ,b ) in L1-norm with the rate λ1 (as computed in (1.17)), i.e.
∞ ∞ 2
I J
kai(t)−ai,∞k2L1(Ω)+ kbj(t)−bj,∞k2L1(Ω) ≤CC−K1P(E(a0,b0)−E(a∞,b∞))e−λ1t
i=1 j=1
X X
where C is the constant in a Csisza´r-Kullback-Pinsker inequality in Lemma 2.2.
CKP
Remark1.1. Theorem1.2isstatedfor renormalisedsolution,whichistheonlyavailable generalconcept
of global solutions for nonlinear reaction-diffusion systems like (1.1)–(1.2).
We remark however that the assumed weak entropy entropy-dissipation law (1.10), which holds on
the formal level, is unclear for renormalised solutions, because of the lacking integrability of the reaction
terms in the entropy dissipation functional (1.16). Hence, our results is formulated for renormalised
solutions, which are supposed to satisfy the weak entropy entropy-dissipation law (1.10).
For classical and weak solutions of (1.1)–(1.2), however, the weak entropy entropy-dissipation law
(1.10) can typically be verified rigorously (even with an equality sign). This was done, for instance, in
[DFPV07] for weak(L logL)2-solutionsof asystemwithquadratic nonlinearities (seealsoTheorem 1.3).
More recently, global weak solutions satisfying (1.10) were shown to exist for systems of the form (1.1)–
(1.2)inallspacedimensionsprovided a(dimension-andnonlinearity-dependent) ”closeness”assumption
onthediffusioncoefficients, seee.g. [FL16]. Imposingastronger”closeness”assumptiononthediffusion
coefficients allows to even show the existence of global classical solutions, see e.g. [FLS16].
CONVERGENCE TO EQUILIBRIUM FOR REACTION-DIFFUSION SYSTEMS 7
Remark 1.2. Theorem 1.2 generalises the previous results of [DF06, DF08, GZ10, MHM15, FL16],
where only special cases of system (1.13) were treated.
Assecondexample(1.12),afterassuming(forthesakeofsimplicity)thatalltheforwardandbackward
reaction rate constants are normalised to one, the reaction-diffusion system modelling (1.12) reads (by
reverting back to the general notation of system (1.1)) as
∂ c d ∆c = c c +c , in Ω R ,
t 1 1 1 1 2 3 +
− − ×
∂ c d ∆c = c c +c , in Ω R ,
 t 2− 2 2 − 1 2 3 × +
∂∂∂tttccc543−−ddd543∆∆∆ccc543 ===−c1ccc442cc+55++c4ccc335,,−2c3, iiinnn ΩΩΩ××RRR+++,,, (1.18)
− − ×
c ν =0, i=1,2,...,5, on ∂Ω R ,
where di > 0, i = 1,...5,∇cair(exi,·p0o)si=tivcei,0d(ixff)u,sion coefficients. The fionurΩm, a×ss c+onservation laws of (1.18)
are
c +c +c =M , i 1,2 and j 4,5 (1.19)
i 3 j i,j
∀ ∈{ } ∀ ∈{ }
andamongthesetherearem=3linearindependentconservationlaws,thusQ R3 5. Inthefollowing,
×
∈
wedenotebyc=(c ,...,c )theconcentrationvectorandby(M )=(M ,M ,M ,M ) R4 the
1 5 i,j 1,4 1,5 2,4 2,5
∈
vector of conservedmasses. Note that the initial mass vector M is determined by any three coordinates
of (M ) R4 corresponding to three linear independent conservation laws and by a given vector
i,j ∈ +
(M ) R4 wemeanthatthesethreecoordinatesaregivenandtheremainingcoordinateissubsequently
i,j ∈ +
calculated. The unique positive detailed balance equilibrium c = (c ,...,c ) R5 to (1.18) is
1, 5,
∞ ∞ ∞ ∈
defined by
c +c +c =M , i 1,2 and j 4,5 ,
i, 3, j, i,j
∞ ∞ ∞ ∀ ∈{ } ∀ ∈{ }
c c =c , (1.20)
 1, 2, 3,
∞ ∞ ∞
c4, c5, =c3, .
∞ ∞ ∞
The corresponding entropy and entropy-dissipationfunctionals for system (1.18) are
5
(c)= (c logc c +1)dx (1.21)
i i i
E −
i=1ZΩ
X
and
5 c 2 c c c c
(c)= d |∇ i| dx+ (c c c )log 1 2 +(c c c )log 4 5 dx, (1.22)
i 1 2 3 4 5 3
D c − c − c
i=1ZΩ i ZΩ(cid:18) 3 3 (cid:19)
X
respectively.
Theorem 1.3 (Explicit convergence to equilibrium for the chain reaction (1.12)).
Let Ω Rn be a bounded domain with smooth boundary. Assume positive diffusion coefficients d > 0
i
⊂
for all i=1,...,5.
Suppose assumptions (A1), i.e. that system (1.18) has a unique positive detailed balance equilibrium
(1.20) and observe that system (1.18) has no boundary equilibria. Note that for a given positive initial
mass vector M R3 corresponding to three linear independent conservation laws (1.19), the assumption
∈ +
(A1) is in fact a consequence of Lemma 4.2 and thus satisfied.
Then, for any nonnegative function c=(c ,...,c ) L1 Ω;[0,+ )5 satisfying the mass conserva-
1 5
∈ ∞
tion laws (1.19), the EED estimate
(cid:0) (cid:1)
(c) λ ( (c) (c )) (1.23)
2
D ≥ E −E ∞
holds for (c) and (c) defined in (1.21) and (1.22) respectively, and where λ >0 is a positive constant
2
E D
which can be explicitly estimated in terms of the initial mass vector M, the domain Ω and the diffusion
coefficients d ,i=1,2,...,5.
i
Moreover, for any nonnegative initial data c Lp(Ω)5 for some p>2 (sufficiently close to 2), there
0
∈
exists a global Lp-weak solution c to (1.18). These weak solutions satisfy the weak entropy entropy-
dissipation law,
t
(c(t))+ (c(s))ds= (c ).
0
E D E
Z0
8 K.FELLNER,B.Q.TANG
Consequently, these weak solutions converge exponentially to the equilibrium, i.e.
5
kci(t)−ci,∞k2L1(Ω) ≤CC−K1P(E(c0)−E(c∞))e−λ2t, ∀t>0,
i=1
X
where λ is in (1.23) and C is the constant in the Csisza´r-Kullback-Pinsker inequality.
2 CKP
The proofs of the above Theorems 1.2 and 1.3 – in particular the corresponding EED estimates – are
based on the following If-Theorem 1.4, which can be understood as a proof of concept of how to derive
explicit EED estimates for general mass-action-law detailed balance reaction-diffusion systems. More
precisely, the following Theorem 1.4 shows that provided the conservation laws of a suitable reaction-
diffusion systems are sufficiently explicitly given to prove two natural key inequalities, then an EED
estimate with explicitly estimable constants and rates follows from a general method of proof.
Theorem 1.4 (Entropyentropy-dissipationestimates forgeneraldetailedbalanceRDnetworks(1.1)).
Let Ω Rn be a bounded domain with smooth boundary. Assume positive diffusion coefficients d > 0
i
for all⊂i = 1,...,I. Assume stoichiometric coefficients αr = (αr,...,αr) ( 0 [1, ))I and βr =
1 I ∈ { }∪ ∞
(βr,...,βr) ( 0 [1, ))I and normalised reaction rate constants kr > 0 as rescaled in (1.4) for all
1 I ∈ { }∪ ∞
r = 1,2,...,R. Suppose assumptions (A1) and (A2), i.e. that system (1.1)–(1.2) features a unique
positive detailed balance equilibrium c of the form (1.7) and no boundary equilibria. The positivity
∞
of the detailed balance equilibrium follows, for instance, from supposing a positive initial mass vector
M Rm corresponding to m linear independent conservation laws (1.3), see e.g. [GGH96, Lemma 3.4].
∈ +
Moreover, assume
i) that for all bounded states c [0,K]I (for a K > 0 sufficiently large), which satisfy the conser-
∈
vation laws
Qc=M,
there exists a constant H >0 such that
4
R αr βr 2 I 2
c c √c
i
H 1 , (1.24)
Xr=1"rc∞ −rc∞ # ≥ 4Xi=1(cid:18)√ci,∞ − (cid:19)
ii) and that there exists 0 < ε 1 such that if c ε2 for some i 1,...,I , then we can find
≪ i0 ≤ 0 ∈ { }
H (ε)>0 depending on ε such that
5
I R αr βr 2
√c 2 + √c √c H . (1.25)
k∇ ikL2(Ω) − ≥ 5
i=1 r=1(cid:18) (cid:19)
X X
Then, for the entropy and entropy-dissipation functional (1.5) and (1.6), the key entropy entropy-
dissipation inequality (1.9), i.e.
(c) λ( (c) (c ))
D ≥ E −E ∞
holds true for all c L1(Ω;[0,+ )I) obeying the mass conservation Qc = M, and where λ =
λ(Ω,H ,H ,D,M,αr,∈βr,kr) is a ∞constant depending explicitly on the domain Ω, the constants H
4 5 4
and H , the diffusion coefficients D, the stoichiometric coefficients αr,βr, the reaction rate constants
5
kr, and the initial mass vector M.
The first assumption (1.24) in Theorem 1.4 can be either interpreted as a quantitative version of
the uniqueness of the positive detailed balance equilibrium (1.7) or an kind of EED estimate (w.r.t.
L2-distances of square-roots of concentrations) for the ODE system associated to (1.1)–(1.2). Indeed,
both viewpoints reflect the observation that the positive detailed balance equilibrium c balances all
reactions(infactthelefthandsideof (1.24)ispreciselyzeroforallstatesc,whichbalanc∞eallreactions)
and satisfies all conservation law Qc = M. These two conditions uniquely define the positive detailed
balanceequilibriumc ,whichistheonlystateforwhichthe righthandsideof (1.24)iszero. Moreover,
consideringboundeds∞tates0 c KconstitutesnorelevantrestrictionsinceLemma2.1belowwillshow
≤ ≤
that all solutions to (1.1)–(1.2) with bounded initial relative entropy satisfy naturally uniform-in-time
bounds 0 c K for some sufficiently large constant K >0.
≤ ≤
Weremarkthatinequality(1.24)canbeshowntoholdwithanexplicitconstantviaTaylorexpansion
for states c near the equilibrium c , yet this is insufficient to prove an explicit bound for the constant
H for states c far from equilibrium∞due to the non-convexity of the problem, see [MHM15].
4
In our proof of the Theorems 1.2 and 1.3, we present a global method of proving inequality (1.24)
with an explicit bound for H . The key to this proof is to exploit the structure of the corresponding
4
CONVERGENCE TO EQUILIBRIUM FOR REACTION-DIFFUSION SYSTEMS 9
conservations laws (1.14) and (1.19), respectively. In fact, for the proof of Theorem 1.2, it is sufficient
to observe that the conservations laws (1.14) entail some qualitative sign relations between the states c
around the equilibrium c .
∞
The second inequality (1.25) is a quantified version of the natural observation that all states c, for
which less than a sufficiently small amountof mass is presentin at leaston concentrationc <ε (recall
i0
thatthe detailedbalanceequilibriumc containsafixedpositiveamountofmassinallconcentrations),
∞
are necessarily far from equilibrium in the sense that left hand side of (1.25), which is a lower bound
for the entropy dissipation D (see Section 2.2), is itself bounded below by a positive constant H (ε).
5
More precisely, (1.25) can be interpretated that such states are either inhomogeneous in space and
thus dissipate entropy in terms of the Fisher information (which is the first term on the left hand side
of (1.25)) or that such states are still subject to a significant amount of chemical reactions and thus
dissipate entropy in terms of the second term on the left hand side of (1.25). The inequality (1.25)
also reflects the fact that the considered system possess no boundary equilibria, and that the entropy
dissipation is bounded away from zero when the concentration is close to the boundary ∂RI .
+
Finally, we remark our believe that our method of proving the inequalities (1.24) and (1.25) in The-
orems 1.2 and 1.3 should also apply to any other systems of the form (1.1)–(1.2), once the structure of
the conservation laws is sufficiently explicitly given. The main reason for having to state Theorem 1.4
as an If-Theorem by assuming the inequalities (1.24) and (1.25) is the fact that we do not know how to
provideanexplicit proofofthese twonaturalinequalities onlybasedonthe pureexistence ofthe matrix
Q, without knowledge of its structure which formalises the conservation laws.
Therestofthis paperis organizedasfollows: InSection2,wefirstprovidepreliminaryestimates and
results before proving Theorem 1.4 for general systems of the form (1.1)–(1.2). The proofs of Theorems
1.2 and 1.3 are presented in Sections 3 and 4 respectively. Finally, we discuss the further possible
applications and some open problems in Section 5.
2. The general method
Inthissection,wefirstbrieflystatepreliminaryestimatesandinequalitiesofthemassactionreaction-
diffusion systems (1.1)–(1.6) before we present the details of our proposed method.
The following notations and elementary inequalities are used in our proof:
L2(Ω)-norm:
For the restof this paper, we will denote by the usual normof L2(Ω): f 2 = f(x)2dx.
k·k k k Ω| |
Spatial averages and square-root abbreviation:
R
For a function f :Ω R, the spatial averageis denoted by (recall Ω =1)
→ | |
f = f(x)dx.
ZΩ
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]
(c) (c )=( (c) (c))+( (c) (c ))
E −E ∞ E −E E −E ∞
I c I c (2.1)
i i
= c log dx+ c log c +c .
i i i i,
Xi=1ZΩ ci Xi=1(cid:18) ci,∞ − ∞(cid:19)
The Poincar´e inequality: For all f H1(Ω), there exists C (Ω)>0 depending only on Ω such that
P
∈
f 2 C f f 2.
P
k∇ k ≥ k − k
The Logarithmic Sobolev inequality: There exists C >0 depending only on Ω such that
LSI
f 2 f
|∇ | dx C flog dx. (2.2)
LSI
f ≥ f
ZΩ ZΩ
An elementary inequality: (a b)(loga logb) 4 √a √b 2.
− − ≥ −
(cid:0) (cid:1)
10 K.FELLNER,B.Q.TANG
An elementary function:
Consider Φ:[0,+ ) [0,+ ) defined as (and continuously extended at z =0,1)
∞ → ∞
zlogz z+1
Φ(z)= − . (2.3)
2
(√z 1)
−
Then, Φ is increasing with limΦ(z)=1 and limΦ(z)=2.
z 0 z 1
→ →
2.1. Preliminary estimates and inequalities.
Lemma 2.1 (L1-bounds). Assume that the initial data c are nonnegative and satisfies (c ) < + .
0 0
E ∞
Then, the non-negative renormalised solutions to (1.1)–(1.4) satisfy
ci(t) L1(Ω) K :=2( (c0)+I) t>0, i=1,2,...,I.
k k ≤ E ∀ ∀
Proof. For global weak or classical solutions of (1.1)–(1.4), integration of (1.6) over (0,t) leads to
(c(t)) (c ), which yields directly the statement of the lemma (see also below). However, for
0
E ≤ E
renormalised solutions, we remark that the lack of regularity prevents direct integration of (1.6). To
overcome this difficulty, we follow the construction of renormalisedsolutions, see e.g. [Fis15, Lemma 1],
and consider for each ε>0 the approximating weak solution cε of the following approximationof (1.1)
R(cε)
∂ cε D∆cε = (2.4)
t − 1+εR(cε)
| |
subject to homogeneousNeumann boundary condition cε ν =0 and initial conditioncε(x,0)=cε(x)
∇ · 0
wherecε isasuitablesmoothapproximationofc . Wecanalsodefinetheentropyandentropy-dissipation
0 0
functionals corresponding to (2.4) in a similar way to (1.5) and (1.6) as
I
ε(cε)= (cεlogcε cε+1)dx (2.5)
E i i − i
i=1ZΩ
X
and
I cε 2 R 1
ε(cε)= d |∇ i| dx+ kr ((cε)αr (cε)βr)(log(cε)αr log(cε)βr)dx (2.6)
D i cε 1+εR(cε) − −
i=1ZΩ i r=1 ZΩ | |
X X
Note that the approximating solutions cε are sufficiently regular to satisfy
d
ε(cε)= ε(cε) 0
dtE −D ≤
and hence
ε(cε)(t) ε(cε) for all t>0.
E ≤E 0
Due to [Fis15, Lemma 2], cε converges to c almost everywhere for (x,t) Ω (0,+ ). Moreover, we
∈ × ∞
recall that cε is a smooth approximation of c . Then, thanks to the Fatou’s lemma, we can pass to the
0 0
limit ε 0 and obtain
→
(c)(t) (c ) for all t>0 (2.7)
0
E ≤E
or equivalently
I
(c (x,t)logc (x,t) c (x,t)+1)dx (c ) for all t>0.
i i i 0
− ≤E
i=1ZΩ
X
By using the elementary inequalities xlogx x+1 (√x 1)2 1x 1 for all x 0, we get
− ≥ − ≥ 2 − ≥
I
1
c (x,t)dx (c )+I.
i 0
2 ≤E
i=1ZΩ
X
This, combined with the non-negativity of solutions, completes the proof of the Lemma. (cid:4)
The following Csisza´r-Kullback-Pinskertype inequality shows that L1(Ω)-convergence to equilibrium
followsfromconvergenceofsolutionsinrelativeentropy (c) (c ). ForageneralisedCsisza´r-Kullback-
E −E ∞
Pinsker inequality, we refer to [AMTU01]. Here, we give an elementary proof using only the natural
bounds given in Lemma 2.1.