Discrete coagulation-fragmentation system with transport and diffusion

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

STÉPHANEBRULL

Discrete coagulation-fragmentation system with transport and diffusion

Tome XVII, n^o3 (2008), p. 439-460.

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Annales de la Facult´e des Sciences de Toulouse Vol. XVII, n^◦3, 2008 pp. 439–460

Discrete coagulation-fragmentation system with transport and diffusion

St´ephane Brull⁽¹⁾

ABSTRACT.—We prove the existence of solutions to two infinite sys- tems of equations obtained by adding a transport term to the classical discrete coagulation-fragmentation system and in a second case by adding transport and spacial diffusion. In both case, the particles have the same velocity as the fluid and in the second case the diffusion coefficients are equal. First a truncated system in size is solved and after we pass to the limit by using compactness properties.

R^´ESUM ´E.—On d´emontre l’existence de solutions pour deux syst`emes infinis d’´equations de coagulation-fragmentation. Dans un premier cas, on rajoute un terme de transport au syst`eme classique de coagulation- fragmentation et dans un second cas on rajoute un terme de transport et un terme de diffusion. Dans les deux cas les particules poss`edent la mˆeme vitesse que le fluide et dans le second cas les coefficients de diffusion sont

´egaux. On r´esout dans un premier temps un probl`eme tronqu´e en taille puis on passe `a la limite en utilisant des lemmes de compacit´e.

1. Introduction

Coagulation and fragmentation processes describe the mecanism by which clusters can coalesce with other clusters to form a larger cluster and can fragment to form two smaller pieces. The clusters are usually identified to their size which can be a positive number in the case of a continous model or an integer in the case of a discrete model. In this paper, only discrete models will be considered.ci(t, x) will denote the concentration of clusters containing i particles (i-mers, denoted by Pi in the sequel) at time t and positionx.iwill be called the size variable. More precisely, the coagulation corresponds to the chemical reaction

Pi+Pj →Pi+j.

(∗) Re¸cu le 2 mars 2007, accept´e le 10 avril 2008

(1) ANLA, University of Toulon, avenue de l’universit´e, 83957 La Garde, France.

Two clusters of sizeiandjwill give a bigger cluster of sizei+j. The velocity of this reaction is equal to ai,jcicj, where (ai,j)_(i,j)∈N∗×N^∗ is called the coagulation kernel. The fragmentation corresponds to the inverse reaction

Pi+j→Pi+Pj.

One cluster containing i+j particles gives two clusters containing respec- tively i and j particles. The velocity of this reaction is equal to bi,jci+j, where (bi,j)(i,j)∈N^∗×N^∗ is called the fragmentation kernel.

When the two equations are in competition, we obtain the equilibrium Pi+Pj−→←Pi+j.

The velocity vi,j of this reaction is vi,j = ai,jcicj −bi,jci+j. As a math- ematical point of view, this problem has been widely studied in ([1], [2], [3]).

In the same time, these particles are in a mouving fluid and the Fick law reads ([9], [1])

∂tci+div(ji) =Qi(c), (1.1) where ji is the flux due to the motion of the clusters of sizei. The flux is decomposed into two terms as ji =uici−di∇ci. ui is the velocity of the clusters of sizeianddi is the velocity of diffusion of the particles of sizei.

The termuici is due to the displacement of the fluid and the termdi∇ci

is due to the diffusion of the particles in the fluid. These two terms depend on the geometry and on the velocity of the flow.

If the fluid is viscous enough, the diffusion term of the flux is prepon- derant compared to the term due to the transport, (1.1) becomes

∂tci−d(i)∆ci=Qi(c).

This case has been investigated in ([6], [7], [10]). An existence theorem is proved in ([7]) when the diffusion coefficients d(i) are equal and in ([6]), the asymptotic behaviour of the solutions in time is studied. In ([11]) an existence theorem is established when the size variable is continous.

When the transport term is preponderant compared to the diffusion term, one obtains

∂tci+div(uci) =Qi(c). (1.2) This case has been studied in ([4], [8]) for continous models.

When both phenomena are comparable, the equation (1.1) reads

∂tci+div(uci)−d(i)∆ci=Qi(c). (1.3) Qi(c) is a source term coming from the velocity of the reaction defined by Qi(c) =

i−1

k=1

ak,i−kckci−k+ 2 ∞ k=1

bk,ici+k−2 ∞ k=1

ak,ickci−

i−1

k=1

bk,i−kci.(1.4)

The term

i−1

k=1

ak,i−kckci−k accounts the formation of clustersPi by coales- cence of two smaller clusters. 2

∞ k=1

bk,ici+k represents the gain of clustersPi

by fragmentation of larger clusters. 2 ∞ k=1

ak,ickci accounts the depletion of clustersPi by coagulation with another cluster, and

i−1

k=1

bk,i−kci represents the fragmentation of clustersPi into two smaller clusters.

In this paper, we will assume that the coefficientsai,k andbi,kfulfill the conditions

ai,k=ak,i>0, bi,k=bk,i >0, ai,k=o(k)bi,k =o(k), sup

k

ai,k

k <∞, sup

k

bi,k

k <∞. (1.5) u(t, x) is the velocity of the fluid in which are the clusters. All the clus- ters are supposed to have the velocity of the fluid which is assumed to be incompressible.

This paper is organized as follows. The second section is devoted to the case 1.2. An existence theorem is proved when all the clusters have the velocity of the fluid. The third section deals with the case 1.3 where an existence theorem is established.

2. The case of pure transport 2.1. Setting of the problem

Consider the equation for any i∈N^∗,

∂tci(t, x) +div(uci)(t, x) =Qi(c)(t, x), t >0, x∈R^D, (2.6)

ci(0, x) =c⁰_i(x), x∈R^D. (2.7)

whereQi has been defined in (1.4). We will consider weak solutions to the problem (2.6-2.7) in the following sense

Definition 2.1. —(ci)i∈N^∗ is a weak solution to the problem (2.6-2.7) if for anyi∈N^∗,ci∈ C⁰(R+×R^D)and

−

R^Dc⁰_i(x)ϕ(0, x)dx−

R⁺

R^Du ci(t, x)∇xϕdsdx

−

R+

R^Dci(t, x)∂tϕdsdx=

R+

R^DQi(c)(t, x)ϕ(t, x)dxds, for each ϕ∈ C¹(R+×R^D) with compact support inR+×R^D.

In this section the main result is,

Theorem 2.2. — Assume that the coefficients ai,k and bi,k satisfy the assumptions (1.5), that the initial data satifies for anyi∈N^∗

c⁰_i 0, c⁰_i ∈C¹(R^D), (i, x) →∂xc⁰_i(x)∈L^∞(R^D×N), ρ⁰=

+∞

i=1

ic⁰_i ∈L^∞(R^D), and that the velocityu∈C¹(R+×R^D)is bounded, fulfills the incompress- ibility conditiondiv(u) = 0, and is such that∂xuis bounded.

Then, the system (2.6-2.7) has a weak solution (ci)i∈N in the sense of Definition 2.1.

As in ([7]-[14]), we shall proceed into two steps. First, a truncated prob- lem in size will be solved by a fix point argument and in a second step we will pass to the limit.

2.2. Resolution of a truncated problem

LetN ∈Nbe given. In this part, the sizes greater thanN are neglected by considering the following problem fori∈ {1· · ·N}.

∂tc^N_i (t, x) +div(uc^N_i )(t, x) = (G^N_i −P_i^N)(c^N)(t, x), t >0, x∈R^D, (2.8)

c^N_i (0;x) =c^0,N_i (x), x∈R^D, (2.9) where

c^0,N_i =c⁰_i if iN c^0,N_i = 0 else

G^N_i (c) =

i−1

k=1

ak,i−kckci−k+ 2

N−i k=1

bk,ici+k and P_i^N(c) =ν_i^N(c)ci (2.10)

with

ν_i^N =

i−1

k=1

bk,i−k+ 2

N−i

k=1

ak,ick,i.

Proposition 2.3. — Under the hypotheses of Theorem 2.2, the system (2.8-2.9) possesses a unique solution on R+×R^D.

The solution of the truncated problem (2.8-2.9) will be the fix point of a mapping Γ. First, the following Lemma whose proof is given in ([7]) will be used.

Lemma 2.4. — For all N-uple{c1...cN} such that, for all i,ci 0, it holds that

νi^N(c)0, G^Ni (c)0, |νi^N|AN sup

i=1..N

ci+BN,

|G^Ni (c)|( sup

i=1..N

c^N_i )²AN +BN sup

i=1..N

c^N_i .

Consider YN the solution to the Cauchy problem dYN

dt =ANY_N²+BNYN (2.11)

YN(0) =R0 t >0 (2.12)

Let TN be the time of existence of the solution and T ∈]0, TN[. Consider the space

E={c∈[C⁰([0;T]×R^D)]^N∩(L^∞([0, T]×R^d))^N;c0;

c(t, .)[L∞(R^d)]N YN(t), t∈[0, T]}. Consider the following iteration

∂tdⁿ⁺¹_i (t, x) +u(t, x).∇xdⁿ⁺¹_i (t, x) = [G^N_i (dⁿ_i)−ν(dⁿ_i)dⁿ⁺¹_i ](t, x), (2.13)

dⁿ⁺¹_i (0, x) =c⁰_i(x).

(2.14) Let Γ be the map defining this iteration.

Lemma 2.5. — For the same assumptions as for Lemma 2.4,E is stable by Γ.

Proof of Lemma 2.5. — The system (2.8-2.9) is solved on a caracteristic.

Consider the vector fieldX(s, t, x) satisfying

∂sX(s, t, x) = u(s, X(s, t, x)), (2.15)

X(t, t, x) = x. (2.16)

Then, the solution to the system (2.8-2.9) writes for anyi∈ {1· · ·N}. c^N_i (s, X(s, t, x)) =c⁰_i(X(0, t, x)) +

t 0

[G^N_i (c^N)−ν_i^Nc^N_i ](s, X(s, t, x))ds.

(2.17) So, by considering the quantity

dⁿ⁺¹_i (τ, X(τ, t, x)) exp

τ

0

ν_i^N(dⁿ)(s, X(s, t, x))ds

, the solution to the system (2.13-2.14) reads for anyi∈ {1· · ·N}

dⁿ⁺¹_i (t, x) =c⁰_i(X(0, t, x)) exp(− t

0

ν_i^N(dⁿ)(s, X(s, t, x))ds +

t 0

exp(−

t s

ν_i^N(dⁿ)(σ, X(σ, t, x))dσ))G^N_i (dⁿ_i(s, X(s, t, x))ds.(2.18) Then, by continuity of dⁿ_i, dⁿ⁺¹_i is also continous. Moreover, the nonneg- ativity of c⁰_i and G^N_i (dⁿ) implies according to (2.18) the nonnegativity of dⁿ⁺¹_i . On the other hand, asν_i^N 0, (2.18) leads to

dⁿ⁺¹_i (t, x)c⁰i(X(0, t, x)) + t

0

G^Ni (dⁿ)(s, X(s, t, x))ds.

But, asdⁿ∈E and by using Lemma 2.4, it holds that dⁿ⁺¹_i (t, x)c⁰_i(X(0, t, x)) +

t 0

d

dsY^N(s)ds.

So, finally, we get thatdⁿ⁺¹_i (t, x)Y^N(t).

We are going to show that this map is a contraction for the norm

|c|= sup

s∈[0,T]

e^−ωNs|c(s,)[L^∞(R^d)]N

where ωN is a constant which shall be chosen big enough so that Γ is a contraction and whose the choice shall be precized during the proof of the following Lemma.

Lemma 2.6. — There is a nonnegative constant ωN depending only on N such that the mappingΓ is a contraction fromE into itself for the norm

| |.

Proof of lemma 2.6. — By considering two consecutive terms of the it- eration (2.13-2.14) and by substracting them, it holds that

∂t[dⁿ⁺¹_i −dⁿ_i](t, x)+u(t, x).∇X(dⁿ⁺¹_i −dⁿ_i)(t, x)+ν^N(dⁿ_i)(t, x)(dⁿ⁺¹_i −dⁿ_i)(t, x)

= [G^N_i (dⁿ)−G^N_i (dⁿ⁻¹)](t, x) +dⁿ_i(t, x)(ν_i^N(dⁿ_i⁻¹)−ν_i^N(dⁿ_i))(t, x).

So, (dⁿ⁺¹_i −dⁿ_i)(t, x) writes (dⁿ⁺¹_i −dⁿ_i)(t, x) =

t 0

[G^N_i (dⁿ)−G^N_i (dⁿ⁻¹)](τ, X(τ, t, x))dτ

+ t

0

dⁿ_i(τ, X(τ, t, x))[ν_i^N(dⁿ⁻¹)−ν_i^N(dⁿ)](τ, X(τ, t, x)) exp(−

t τ

ν_i^N(dⁿ)(s, X(s, t, x))dsdτ.

According to the expression of G^N_i and P_i^N, there exists a nonnegative constantC(N, T) such that

|G^Ni (c)−G^N_i (d)|C(N, T)c−d[L^∞(Ω)]^N,

|Pi^N(c)−P_i^N(d)|C(N, T)c−d[L^∞(Ω)]^N. (2.19) So,

|(dⁿ⁺¹i −dⁿi)|C(N, T)|dⁿi −dⁿ_i⁻¹|[1−e^−ωNt ωN

].

Hence, by choosingωN big enough such thatk=^2C(N,T)_ω

N <1, the result is

proved.

For the proof of Proposition 2.3, the following Lemma will be used

Lemma 2.7. — ForG^N_i andP_i^N defined in (2.10), it holds that N

i=1

[iG^N_i (c)−

iP_i^N(c)] = 0.

For the proof, see ([7]).

Proof of Proposition 2.3. — From previously, (dⁿ_i)_n∈N is a converging sequence inE and Γ is a contraction. Therefore the problem (2.8-2.9) has a unique solution on [0, T]×R^d forT∈]0;TN[.

In order to get a global solution onR+×R^d, consider ρ^N =

N

i=1

ic^N_i , ρ=

+∞

i=1

ici (2.20)

which represents total mass of particles which react.

Multiply in (2.8-2.9), the equation numberibyi, sum oniuntilN and use Lemma 2.7, leads to

∂tρ^N(t, x) +u(t, x).∇xρ^N(t, x) = 0, (2.21) ρ^N(0, x) =ρ^N0(x) =

N i=1

ic⁰i(x). (2.22)

So, by considering the vector field X(s, t, x) defined by the system (2.15- 2.16), it holds that

0ρ^N(t, x) =ρ^N₀(X(0, t, x))ρ0(X(0, t, x))ρ⁰L^∞(R^D). (2.23) Hence, (∀i∈ {1;N}), c^Ni ρ⁰L^∞(R^D).Next, we chooseR0=ρ0L^∞(R^D)

and we solve the equation (2.6) for any i ∈ {1....N} on [T,2T] with the Cauchy data equal toci(T,·). Then, a reiteration of this process gives global existence of the solution onR+×R^D.

2.3. Solution of the problem

The aim is now to pass to the limit whenN tends to +∞in the system (2.8-2.9).

Proposition 2.8. — For any compact set [0;T]×K of R+×R^D and for anyi∈N, the sequence(c^N_i )N∈Nis strongly compact inC⁰([0;T]×K).

Proof of Proposition 2.8. — In order to apply the Ascoli theorem, we shall control ∂tc^N_i and ∂xc^N_i . Consider BR the closed ball of R^D with a radius equal to R > 0. Hence, by differentiating the relation (2.17) with respect to the space variablexj, it holds that

∂x_j(c^N_i )(t, x) = ∂X[c⁰_i(X(0, t, x))]

+ t

0

∂xj[Q^N_i (c)](s, X(s, t, x))∂xjX(s, t, x)ds. (2.24) Now, in order to estimate ∂tX(s, t, x) and ∂xX(s, t, x), let us show the following Lemma.

Lemma 2.9. — Ifu∈C¹is bounded on[0;T]×R^Dand if∂xuis bounded on [0;T]×R^D then∂xX and∂tX are bounded on[0;T]²×R^D.

Proof of Lemma 2.9. — By integrating (2.15),X writes X(s, t, x) =x+

s t

u(σ, X(σ, t, x))dσ. (2.25) So by differentiating (2.25) with respect to the space variablexand by using that ∂xuis bounded on [0, T]×R^D, there existsM >0 such that

|∂xX(s, t, x)|1 +M( t

s

|∂xX(σ, t, x)|dσ).

So, according to the Gronwall lemma, ∂xX(s, t, x) is bounded on [0;T]²× R^D. Analogously, the same result holds for∂tX(s, t, x).

End of the proof Proposition 2.8. — In order to control the term∂x_jQi

consider the quantity

∂x[G^N_i (c^N)] =

i−1

k=1

ak,i−k

∂xc^N_k c^N_i−k+c^N_k ∂xc^N_i−k

+ 2 N k=1

bk,i∂xc^N_i+k. (2.26)

A bound onc^N_i independent ofN is first researched inL^∞((0, T);R^D).

Asρ^N₀ = N i=1

ic⁰_i andρ0= ∞

i=1

ic⁰_i, withρ0∈L^∞(R^D), it holds that

c^N_i ρ^N0 L^∞(R^D)ρ0L^∞(R^D)M0. (2.27)

Now, let us control the term

i−1

k=1

ak,i−k∂xc^Nk(t, x)c^Ni−k(t, x) of the right-hand side of the equation (2.26).

i−1

k=1

ak,i−k∂xc^Nk(t, x)c^Ni−k(t, x)

i−1

k=1

ak,i−k

i−k |∂xc^Nk(t, x)|(i−k)c^Ni−k(t, x) From assumption (1.5), there is A >0 such that

sup

k∈{1,i−1}|ak,i−k

i−k|A.

Hence,

i−1

k=1

ak,i−k∂xc^Nk(t, x)c^Ni−k(t, x)Asup

i∈N sup

x∈R^D|∂xc^Nk(t, x)|

i−1

k=1

(i−k)c^Ni−k. But, by definition ofρ^N, it comes that

|

i−1

k=1

(i−k)c^N_i−k|ρ^N. Then (2.23) leads to

|

i−1

k=1

(i−k)c^N_i−k|ρL^∞(R^D). Therefore, by using (2.27), we get the estimate

i−1

k=1

ak,i−k∂xc^N_k(t, x)c^N_i−k(t, x)AM0

sup

i∈N sup

x∈R^D|∂xc^N_i (t, x)| .

whereM0is a nonnegative constant. In the same way, we obtain

|

i−1

k=1

ak,i−k∂xc^N_i−k(t, x)c^N_k (t, x)|AM0(sup

i∈N sup

x∈R^D|∂xc^N_i (t, x)|),

|

N−1

k=1

bk,i∂x(c^N_i+k(t, x))|B(sup

i∈N sup

x∈R^D|∂xc^N_i (t, x)|),

whereBis a constant independent of the variablesk, i, t, x. So finally, there exists a constantC independent of the variablest, x, i, N such that

|∂x[G^N_i (c^N)](t, x)|C(sup

i∈N sup

x∈R^D|∂xc^N_i (t, x)|).

The same result holds forP_i^N(c^N). So, sup

x∈R^D∂x[c^N_i (t, x)]L^∞(R^D)C+α t

0

sup

x∈R^D∂x[c^N_i (s, x)]L^∞(R^D)ds, where α and C are nonnegative constants independent of the variables t, x, N. From the Gronwall lemma, it holds that

(∀i∈N^∗),(∀t∈[0;T]),∂xc^N_i (t, x)L∞(R^D)Ce^αt.

By reasonning in the same way, we can prove that ∂tc^N_i is also controled.

Then, by using the Ascoli Theorem, the sequence (c^N_i )N∈Nis strongly com- pact inC⁰([0;T]×BR) for anyi.

Proof of Theorem 2.2. — By using a diagonal process, there is a subse- quence c^φ(N) of c^N such that (∀i∈N), c^φ(N_i ⁾ is converging to a continous functionci uniformly on all compact set of the form [0;T]×BR. By arguing as in ([7]), it holds that G^N_i (c^N) (resp.P_i^N(c^N)) converges toGi(c) (resp.

Pi(c)) uniformly on [0;T]×BR. So, we can pass to the limit in the weak form of (2.6-2.7).

3. The case of transport and diffusion 3.1. Setting of the problem

In this section, consider the problem

∂tci(t, x) +div(uci)(t, x)−∆ci(t, x) =Qi(c)(t, x) (3.28)

ci(0;x) =c⁰_i(x), x∈Ω, (3.29)

∂ci

∂η(t, σ) = 0 t >0, σ∈∂Ω, (3.30)

where Ω is an open bounded set of class C¹ with ∂Ω as boundary and whereQ(c)ihas been defined in (1.4). It corresponds to the case (1.3) when the diffusion coefficients d(i) are taken equal to one. The solutions of the problem (3.28-3.29-3.30) will be considered in the following sense.

Definition 3.1. — (ci)_i∈Nis a weak solution to the problem (3.28-3.29- 3.30) if for anyi∈N^∗ and anyT >0,ci∈L²([0, T]×Ω)and

−

R^Dc⁰_i(x)ϕ(0, x)dx−

R⁺

R^Dci(t, x)∂tϕ(t, x)dx

−

R⁺

R^Dci(s, x)u· ∇xϕ(s, x)dxds−

R⁺

R^D∇xci(s, x)· ∇xϕ(s, x)dxds

=

R+

R^DQi(c)ϕ(s, x)dxds for allϕ∈ C¹(R+×R^D)with compact support in R+×R^D.

The main result of this part is

Theorem 3.2. — Let Ω be an open and bounded set of class C¹. As- sume that the kinetic coefficients (a)_(i,j)∈N∗×N^∗ and (b)_(i,j)∈N∗×N^∗ satisfy the assumptions (1.5), that the initial data satisfies

c⁰_i 0, i∈N^∗, c⁰_i ∈L²(R+×Ω), ρ⁰=

+∞

i=1

ic⁰_i ∈L^∞(Ω).

and that the velocity of the fluid u∈ H¹(R+;H₀¹(Ω)∩L^∞(Ω)) fulfills the incompressibility condition div(u) = 0.

Therefore the system (3.28-3.29-3.30) has a weak solution onR+×R^D in the sense of Definition 3.1.

As previously, we shall proceed into two parts. First, we shall solve a truncated problem and after we will pass to the limit.

3.2. Resolution of a truncated problem

LetN ∈Nbe given. The sizes greater thanN are removed, by consid- ering the following problem for anyi∈ {1· · ·N},

∂tc^N_i +u· ∇xc^N_i −∆c^N_i =Qi(c^N), (3.31) c^N_i (0;x) =c^0,N_i (x) i∈N, t >0, x∈Ω, (3.32)

∂

∂ηc^N_i (t, σ) = 0, t >0, σ∈∂Ω. (3.33)

Proposition 3.3. — Under the assumptions of Theorem 3.2, the prob- lem (3.31-3.32-3.33) has a unique solution defined onR+×R^D.

Consider the following iteration for anyi∈ {1· · ·N},

∂tdⁿ⁺¹_i −∆dⁿ⁺¹_i +u(t, x)· ∇xdⁿ⁺¹_i +ν_i^N(dⁿ_i)dⁿ⁺¹_i =G^N_i (dⁿ), (3.34)

dⁿ⁺¹_i (0;x) =c⁰_i(x), (3.35)

∂

∂ηdⁿ⁺¹_i (t, σ) = 0, t >0, σ∈∂Ω. (3.36) LetS be the mapping defining this iteration.

Lemma 3.4. — If for any i∈ {1...N},ci0, then S(c)i0.

Proof of Lemma 3.4. — Consider ci 0 and put di = S(c)i. Consider f ∈C¹(R) such thatf is nondecreasing on ]0; +∞[ andf(t) = 0 fort∈R−. Multiply the equation (3.34) satisfied by di by f(−di) and integrate on [0;t]×Ω leads to (∀t∈[0;T]),

t 0

Ω

(∂tdi)f(−di)dxdt− t

0

Ω

∆dif(−di)dxds+

t 0

Ω

u·∇xdif(−di)dxdt

+ t

0

Ω

νi^N(c)dif(−di) = t

0

Ω

G^Ni (c)f(−di)dxds. (3.37) But, from the Green formula and by using (3.36), it holds that

− t

0

Ω

∆dif(−di)dxdt= t

0

Ω

∇xdi· ∇xf(−di)dxdt.

Let F be the primitive function of f such thatF(0) = 0. So,F is nonde- creasing on ]0; +∞[ and is identically 0 on ]− ∞; 0]. Hence, we get

t 0

Ω

uj(s, x)∂xjdi(s, x)f(−di)(s, x)dxds=− t

0

Ω

uj(s, x)∂xjF(−di)(s, x)dsdx.

So, by using the Green formula, it comes that

Ω

uj(s, x)∂xjdi(s, x)f(−di)(s, x)dxds = −

Γ

uj(s, σ)F(−di)(s, σ)dσ +

Ω

∂uj

∂xj

F(−di)(s, x)dx.

But, asu∈H0¹(Ω) anddiv(u) = 0, we get t

0

Ω

u· ∇xdif(−di)dxdt= 0.

So, (3.37) gives the inequality,

Ω

F(−di)(x, v)dx−

Ω

F(−di)(0, v)dx+ t

0

Ω

f(−di)|∇xdi|²dxdt +

t 0

Ω

ν_i^N(−di)(s, x) (−d)f(−di)(s, x)dxds0. (3.38)

But, asdi(0;x) =c⁰_i(x),F is identically 0 on ]− ∞; 0] andF(−d)(0;x) = 0.

Moreover,ν_i^N(c) andK(x) =xf(x) are nonnegative quantities. So, t

0

Ω

ν^N(−d)(s, x)(−d)f(−d)(s, x)dxds0.

f being nondecreasingf(−di)0, we get from (3.38),

Ω

F(−di)(t, x)dx0.

F(−di) being nonnegative, we obtain thatF(−di) is zeroa.e.F(−di) being continous, F(−di) is zero on [0;T]×R^D. So, by definition ofF, d0 on [0;T]×Ω.

A bound inL^∞ on the sequencedⁿ_i is now researched. It is given by the following lemma.

Lemma 3.5. — Letd=S(c). Then for any i∈ {1, N},di satisfies di(t,·)L∞(Ω) c⁰_iL∞(Ω)+

t 0

[AN(c(s,·)L^∞(Ω)^N)²

+ BNc(s,·)L∞(Ω)^N]ds. (3.39) Proof of lemma 3.5. — Let us put

f(t) = t

0

[AN(c(s,·)L^∞(Ω)^N)²+BNc(s,·)L^∞(Ω)^N]ds.

Consider now a function Gsuch that G ∈ C¹(R), G is nondecreasing on ]0; +∞[, (∀s0),G(s) = 0. Let us putKi=c⁰iL^∞(Ω),H(s) = ₀^sG(σ)dσ.

For anyi∈ {1, N}, introduce the functionϕi defined by ϕi(t) =

Ω

H(di(t, x)−Ki− t

0

f(s)ds)dx.

For any i ∈ {1, N} ϕi satisfies ϕi(0) = 0, ϕi ∈ C¹(]0; +∞[;R) ϕi 0.

Deriveϕi with respect to the time variablet leads to ϕ_i(t) =

Ω

G[di(t, x)−Ki− t

0

f(s)ds](∂tdi(t, x)−f(t))dx.

Asν_i^N(dⁿ_i)dⁿ⁺¹_i 0, it holds that

∂tdi(t, x)−f(t)∆di(t, x)− D j=1

uj(t, x)∂xjdi(t, x).

So, we get ϕ_i(t)

Ω

G(di(t, x)−Ki− t

0

f(s)ds)∆di(t, x)dx

− D j=1

Ω

uj(t, x)∂xjdi(t, x)G(di(t, x)−Ki− t

0

f(s)ds)dx.(3.40)

The definition ofH implies that

∂x_jH

di(t, x)−K− t

0

f(s)ds

=∂x_jdi(t, x)G

di(t, x)−K− t

0

f(s)ds . By using the Green formula, u∈H₀¹(Ω) anddiv(u) = 0, it comes that

D

j=1

Ω

uj(t, x)∂x_jH[di(t, x)−K− t

0

f(s)ds]dx= 0.

Moreover from the Green formula, it holds that

Ω

G(di(t, x)−Ki− t

0

f(s)ds)∆di(t, x)dx

=

Ω

|∇xdi(t, x)|²G

di(t, x)−K− t

0

f(s)ds dx.

So,G being nondecreasing, (3.40) leads to ϕ_i(t)−

Ω

|∇xdi(t, x)|²G((di(t, x)−K− t

0

f(s)ds)dx <0.

Hence, ϕi is nonincreasing on R+. As, ϕi is nonnegative and satisfies ϕi(0) = 0, ϕi yields 0 everywhere. So, by definition of H, (3.39) holds.

We shall use an analogous method as in the previous part. Consider the Cauchy problem

dYN

dt =ANY_N² +BNYN, (3.41) YN(0) =R0, t >0. (3.42) LetTN be the time of existence ofYN and consider T∈[0;TN]. Define the space

E={c∈(L^∞([0, T]×R^d))^N;c0;c(t, .)[L∞(R^d)]N YN(t), t∈[0, T]}.

Lemma 3.6. — E is stable byS.

Proof of Lemma 3.6. — Let c∈E and d=S(c). From Lemma 3.4, for any i ∈ {1...N}, di 0. As c ∈ E, for any i ∈ {1, N}, ci(t, x) YN(t).

Then, from the maximum principle applied to the equation (3.34), it holds that

di(t,·)L^∞(Ω)c⁰_iL^∞(Ω)+ t

0

[AN(c(s,·)L∞(Ω)^N)²+BNc(s,·)L∞(Ω)^N]ds.

Asc∈E, the equation (3.41) leads to di(t,·)L^∞(Ω)c⁰iL^∞(Ω)+

t 0

dYN(t) dt (s)ds.

By choosingR0>c⁰iL^∞(Ω), the result follows.

Eis equipped with the norm

|c|= sup

s∈[0;T]

|e^−ωNsc(s,·)[L²(Ω)]^N

and the constantωN will be chosen so thatSis a contraction for this norm.

Lemma 3.7. — There is a constant ωN depending only onN such that S is a contraction from E into itself for the norm | |.

Proof Lemma 3.7. — Substract two consecutive terms of the iteration, multiply the last equation by (dⁿ⁺¹_i −dⁿ_i)(t, x) and integrate on [0, T]×Ω leads to

t 0

Ω

(dⁿ⁺¹_i −dⁿ_i)∂t(dⁿ⁺¹_i −dⁿ_i)dxds− t

0

Ω

(dⁿ⁺¹_i −dⁿ_i)∆(dⁿ⁺¹_i −dⁿ_i)dxds +

t 0

Ω

(dⁿ⁺¹_i −dⁿi)u· ∇x(dⁿ⁺¹_i −dⁿi)dxds+ t

0

Ω

ν_i^N(dⁿ_i)(dⁿ⁺¹_i −dⁿi)dxds

= t

0

Ω

(dⁿ⁺¹_i −dⁿi)[G^N_i (dⁿ_i)−G^Ni (dⁿ_i⁻¹)]dxds +

t 0

Ω

dⁿ_i (dⁿ⁺¹_i −dⁿi)[ν^N(dⁿ⁻¹)−ν^N(dⁿ))]dxds. (3.43) But, asdⁿ⁺¹_i (0, x) =dⁿ_i(0, x) =c⁰_i(x), it holds that

t 0

Ω

(dⁿ⁺¹_i −dⁿi)∂t(dⁿ⁺¹_i −dⁿi)(s, x)dxds= 1 2

Ω

(dⁿ⁺¹_i −dⁿi)²(t, x)dx.