Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

J.A. Cañizo

, L. Desvillettes

, K. Fellner

aDepartament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain bCMLA, ENS Cachan, IUF & CNRS, PRES UniverSud, 61 Av. du Pdt. Wilson, 94235 Cachan Cedex, France

cDAMTP, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Received 29 June 2009; received in revised form 26 September 2009; accepted 26 September 2009

Available online 14 October 2009

Abstract

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a globalL²bound on the mass density and was previously used, for instance, in the context of reaction–

diffusion equations.

In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:35B45; 35Q72; 82D60

Keywords:Discrete coagulation–fragmentation systems; Mass conservation; Duality arguments

1. Introduction

We consider the time evolution of a physical system where a set of particles can aggregate into groups of two or more, calledclusters, and where these clusters can diffuse in space with a diffusion constant which depends on their size. If we represent space by an open bounded setΩ⊆R^Nwith regular boundary, the initial–boundary problem for the concentrationsc_i=c_i(t, x)0 of clusters with integer sizei1 at positionx∈Ωand timet0 is given by the discrete coagulation–fragmentation system of equations with spatial diffusion and homogeneous Neumann boundary conditions:

* Corresponding author.

E-mail addresses:[email protected] (J.A. Cañizo), [email protected] (L. Desvillettes), [email protected], [email protected] (K. Fellner).

1 On leave from: Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien, Austria.

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.10.001

∂_tc_i−d_ixc_i=Q_i+F_i forx∈Ω, t0, i∈N^∗, (1a)

∇xc_i·n=0 forx∈∂Ω, t0, i∈N^∗, (1b)

c_i(0, x)=c_i⁰(x) forx∈Ω, i∈N^∗, (1c)

wheren=n(x)represents a unit normal vector at a pointx∈∂Ω,d_i is the diffusion constant for clusters of sizei, and

Q_i≡Q_i[c] :=Q⁺_i −Q⁻_i :=1 2

i−1

j=1

a_i−j,jc_i−jc_j−^∞

j=1

a_i,jc_ic_j,

F_i≡F_i[c] :=F_i⁺−F_i⁻:=^∞

j=1

B_i+jβ_i+j,ic_i+j−B_ic_i. (2)

The parametersB_i,β_i,janda_i,j, for integersi, j0, represent the total rateB_i of fragmentation of clusters of sizei, the average numberβ_i,j of clusters of sizej produced due to fragmentation of a cluster of sizei, and the coagulation ratea_i,j of clusters of sizeiwith clusters of sizej. We refer to these parameters asthe coefficientsof the system of equations. They represent rates, so they are always nonnegative; single particles do not fragment further, and mass should be conserved when a cluster fragments into smaller pieces, so one always imposes

ai,j=aj,i0, βi,j 0

i, j∈N^∗

, (3a)

B1=0, Bi0 i∈N^∗

, (3b)

i=

i−1

j=1

jβ_i,j (i∈N, i2). (3c)

In fact, the last condition (3c) implies the conservation of the total mass

Ω

_∞

i=1ic_idx, which becomes obvious from the following formal fundamental identity or weak formulationof the coagulation and fragmentation opera- tors: Consider a sequence of nonnegative numbers{ci}, and defineQi,Fi as in Eqs. (2), then, for any sequence of numbersϕi,

∞ i=1

ϕ_iQ_i=1 2

∞ i=1

∞ j=1

a_i,jc_ic_j(ϕ_i+j−ϕ_i−ϕ_j), ∞

i=1

ϕiFi= − ∞ i=2

Bici

ϕi−

i−1

j=1

βi,jϕj

. (4)

As a (still formal) consequence for solutions{ci}of (1)–(2), one can calculate the time derivative of the integral of the

moment

ϕici to obtain d

dt

Ω

∞ i=1

ϕ_ic_i=

Ω

∞ i=1

ϕ_i(Q_i+F_i), (5)

since the integral of the diffusion part vanishes due to the homogeneous Neumann boundary condition. By choosing ϕ_i:=iabove and thanks to (3c), we have_∞

i=1iQ_i=_∞

i=1iF_i=0, and the total mass is formally conserved:

ρ(t,·)

L¹=

Ω

∞ i=1

ic_i(t, x) dx=

Ω

∞ i=1

ic⁰_i(x) dx=ρ⁰

L¹ (t0). (6)

Our main aim in this work is to provide some new bounds on the regularity of weak solutions for system (1)–(2) by means of techniques developed in the context of reaction–diffusion equations [9,16,17], and to give three applications to those bounds, the main one proving rigorously (for almost all the coefficients where this is true in the homogeneous case) mass conservation (6) and thus the absence of gelation, a well-known phenomenon in coagulation–fragmentation models [11,10], where the formal conservation of mass is violated as clusters of infinite size are formed.

In this paper we will work with the global weak solutions constructed in [15] (see also [20] for the caseΩ=R^N) under the assumption

j→+∞lim a_i,j

j = lim

j→+∞

B_i+jβ_i+j,i

i+j =0 (for fixedi1). (7)

The notion of solution is the following, which we take from [15]:

Definition 1.1.A global weak solutionc= {ci}i1to (1)–(2) is a sequence of functionsci: [0,+∞)×Ω→ [0,+∞) such that for eachT >0,

c_i∈C

[0, T];L¹(Ω)

, i1, (8)

∞ j=1

a_i,jc_ic_j∈L¹

[0, T] ×Ω

, (9)

sup

t0 Ω

_∞

i=1

ic_i(t, x)

dx

Ω

_∞

i=1

ic⁰_i(x)

dx, (10)

and for eachi1,ci is a mild solution to thei-th equation in (1a), that is, c_i(t )=e^diA1tc⁰_i +

t

0

e^diA1(t−s) Q_i

c(s) +F_i

c(s)

ds, t0, (11)

whereQi[c]is defined by (2),A1denotes the closure inL¹(Ω)of the unbounded linear operatorAofL²(Ω)defined by

D(A):=

w∈H²(Ω)∇w·n=0 on∂Ω

, Aw=w, (12)

ande^diA1tis theC₀-semigroup generated byd_iA₁inL¹(Ω).

The existence result of [15] reads:

Theorem 1.2(Laurençot–Mischler). Assume hypotheses(3)and(7)on the coagulation and fragmentation coefficients.

Assume also that

di>0 for alli1,

and that the nonnegative initial datum has finite mass:

c⁰_i 0 onΩ and

Ω

∞ i=1

ic_i⁰<+∞.

Then, there exists a global weak solution to the initial–boundary problem(1)–(2)in the sense of Definition1.1.

Under the extra assumptions on the diffusion constants and the initial data 0<inf

i {d_i} =:d, D:=sup

i

{d_i}<+∞, (13)

∞ i=1

ic_i⁰∈L²(Ω), (14)

we are in fact able to prove the followingL²bound on the mass densityρ(t, x):=_∞

i=1ic_i(t, x): Denoting byΩ_T the cylinder[0, T] ×Ω, we have

Proposition 1.3.Assume that(3), (7), (13)and(14)hold. Then, for allT >0the massρof a weak solution to system (1)–(2) (given by Theorem1.2)lies inL²(Ω_T)and the following estimate holds:

ρ _L2(ΩT)

1+sup_i{d_i} infi{d_i}

Tρ(0,·)

L²(Ω). (15)

Remark 1.4.Note that the assumption (7) is only included in Proposition 1.3 in order to ensure the existence of a weak solution via Theorem 1.2. Without assumption (7), the bound (15) would still hold for smooth solutions of a truncated version of system (1)–(2) uniformly with respect to the truncation. See [15] for the details of such a truncation.

In addition to Proposition 1.3, we give a new proof of anL¹bound of the various coagulation and fragmentation terms:

Proposition 1.5.We still assume that(3), (7), (13)and(14)hold. Then, for allT >0andi∈N^∗all the termsQ⁺_i , Q⁻_i ,F_i⁺andF_i⁻associated to a weak solution to system(1)–(2) (given by Theorem1.2)lie inL¹(ΩT)with a bound which depends in an explicit way on the coagulation and fragmentation coefficients, the diffusion coefficients, and the initial datac⁰_i.

Remark 1.6.The fact that the termsQ⁺_i ,Q⁻_i ,F_i⁺andF_i⁻associated to a weak solution are inL¹(Ω_T)is included in the definition of weak solution; the main content of Proposition 1.5 is the explicit dependence of the bounds on the coefficients and initial data, which can be used to obtain uniform estimates for approximated solutions as we show for instance in Section 3. For details on the explicitL¹bounds we refer to the proof of Proposition 1.5 in Section 2.

Remark 1.7.The L¹ bounds onQ⁺_i ,Q⁻_i ,F_i⁺ andF_i⁻require the assumption (7) only to ensure existence. They would hold at the formal level (that is, for smooth solutions of a truncated system) under the less stringent assumption

K_i:=sup

j∈N

Bi+jβi+j,i

i+j <+∞

i∈N^∗

. (16)

Note that the aboveL¹bound also holds when assumptions (3), (7) are replaced by the assumptions of Theorem 1.2 in [15], but the proof is then much more difficult as it requires an induction oniwhich can be removed under our extra assumptions.

In Section 3, as a first application of the bounds obtained in Propositions 1.3 and 1.5, we give a very simple proof of existence of weak solutions to (1)–(2) in dimensionN=1 (that is, the result of Theorem 1.2 in dimension 1) under the additional assumptions (13) and (14).

Our main application of Propositions 1.3 and 1.5 is however related to the problem of conservation of mass (6), which holds rigorously for solutions to a truncated system (see e.g [15]). Nevertheless, it is an important issue in coagulation–fragmentation theory whether (6) holds for weak solutions of system (1)–(2) itself, or if (6) is replaced by an inequality stating that mass is nonincreasing in time. If at some time t, the identity (6) does not hold any more, we say that gelation occurs, which means from a physical point of view that a macroscopic object has been created.

Our main result in Section 4 basically shows that (under the assumptions (3) and (7)) gelation does not occur when the coagulation coefficientsa_i,j are at most linear and, moreover, slightly sublinear far off the diagonali=j. More precisely, we prove mass conservation under the following condition on the coefficientsa_i,j:

Hypothesis 1.8.There is some bounded functionθ: [0,+∞)→(0,+∞)such thatθ (x)→0 whenx→ +∞and

a_i,j(i+j )θ (j/ i) for allji. (17)

(Or equivalently, by symmetry, ai,j(i+j )θ

max{j/ i, i/j}

for alli, j1.)

Theorem 1.9.Assume that(3), (7), (13), and(14)hold. Also, assume Hypothesis1.8. Then, the weak solution to the system(1)given by Theorem1.2has a superlinear moment which is bounded on bounded time intervals;this is, there is some increasing functionC=C(T ) >0, and some increasing sequence of positive numbers{ψ_i}i1with

ilim→∞ψ_i→ +∞ (18)

such that for allT >0,

Ω

∞ i=1

iψ_ic_iC(T ) for allt∈ [0, T]. (19)

As a consequence, under these conditions all weak solutions given by Theorem1.2of (1)conserve mass:

Ω

ρ0(x) dx=

Ω

ρ(t, x) dx for allt0. (20)

Remark 1.10(Admissible coagulation coefficients).Let us comment on Hypothesis 1.8. First note that 1.8 includes coefficients of the form

a_i,j Cst

i^αj^β+i^βj^α

for anyα, β >0 such thatα+β1 (takeθ (x)=x^−εforε >0 small enough). It is also satisfied when ai,j Cst

i φ(i)+ j

φ(j )

,

wherex→φ(x)is any positive strictly increasing function (forx big enough), which goes to infinity at infinity, and such that x→ _{φ (x)}^x is also increasing (take θ (λ)=φ(λ)^−1/2). All the examplesφ=log(1+ ·),φ=log(1+ ·)◦ log(1+ ·), . . . ,φ=log(1+ ·)◦ · · · ◦log(1+ ·)satisfy this condition. Likewise, condition (17) also holds when (for i, j2)

a_ijCst

iR(logj )

logi +jR(logi) logj

(21) for some nondecreasing functionR such that x→R(x)/x is nonincreasing and tends to 0 whenx→ +∞. Note indeed that when (21) holds,

a_ij

i+j 1 1+j/ i

R[log(j/ i)+logi]

logi + j/ i

1+j/ i

R[logi]

log(j/ i)+logi. (22)

Then, condition (17) is obtained by distinguishing the casesij/ iandij/ iin both terms of the right-hand side of (22).

Assumption (21) can even be replaced by a_ijCst

iR(log(logj ))

log(logi) +jR(log(logi)) log(logj )

,

with the same requirements onRas previously.

Note however that the linear coefficientaij=i+j(or the coefficientaij=_logiⁱ logj+_logj^j logi) does not satisfy Hypothesis 1.8, though one would expect that Theorem 1.9 still holds for such coefficients.

Before introducing a generalised coagulation–fragmentation model and thus, a third application of Propositions 1.3 and 1.5, let us briefly review previous results on existence theory and mass conservation for the coagulation–

fragmentation system (1). With some further restrictions on the coefficients as compared to [15], existence of solutions by means ofL^∞bounds on thec_i has been proven in [3,7,13,18,19]. A different technique was used in [1] to prove that Eq. (1) is well posed, locally in time, and globally in time when the space dimensionN is one, always assuming that the coagulation and fragmentation coefficients are bounded.

In a recent work [14], Hammond and Rezakhanlou considered Eq. (1) without fragmentation, and gaveL^∞bounds on moments of the solution (and as a consequence,L^∞bounds on thec_i). This implies uniqueness and mass conser- vation for some coagulation coefficients that grow at most linearly as well as an alternative proof of the existence of L^∞solutions by a priori bounds on thec_i; for instance, ifΩ=R^N and diffusion coefficientsd_i are nonincreasing and satisfying (13) and if moreover

∞ i=1

ic⁰_i ∈L^∞ R^N

,

∞ i=1

i²c⁰_i ∈L¹ R^N

, a_i,j C(i+j )

for some C >0 and alli, j 1, then they show that mass is conserved for all weak solutions of Eq. (1) without fragmentation. See [14, Theorems 1.3 and 1.4] and [14, Corollary 1.1] for more details.

In the spatially homogeneous case, mass conservation is known for general data with finite mass and coagulation coefficients including the critical linear casea_i,jCst(i+j )(see, for instance, [2,5]).

We finally give a third application of Propositions 1.3 and 1.5. As mentioned already in Remarks 1.4 and 1.7, Propo- sitions 1.3 and 1.5 (despite true without restrictions on the coagulation coefficients a_i,j for smooth approximating solutions) do not really improve the theory of existence of weak solutions for the usual models of coagulation–

fragmentation like (1) as the full assumption (7) are needed in passing to the limit in the approximating solutions. At best they help provide simpler proofs in particular cases, as done in Section 3.

On the other hand, Propositions 1.3 and 1.5 are well suited for the existence theory of more exotic models, for instance, when fragmentation occurs due to binary collisions between clusters. Then, the break-up terms are quadratic, being proportional to the concentration of the two clusters which collide. This leads to coagulation–fragmentation models where all terms in the right-hand side are quadratic.

More precisely, we consider that clusters of sizekandlcollide with a rateb_k,l0, leading to fragmentation. As a consequence, clusters of sizei <max{k, l}are produced, in average, at a rateβ_i,k,l0 in such a way that the mass is conserved (that is,

i<max{k,l}iβ_i,k,l=k+l). This leads to the following system (fort ∈R₊,x∈Ω a bounded regular open subset ofR^N):

∂_tc_i−d_ixc_i=1 2

k+l=i

a_k,lc_kc_l−^∞

k=1

a_i,kc_ic_k+1 2

∞ k,l=1

i<max{k,l}

b_k,lc_kc_lβ_i,k,l−^∞

k=1

b_i,kc_ic_k i∈N^∗

,

(23) together with the initial and boundary conditions (1b), (1c). For this model, the set of assumptions (3) is replaced by

ai,j=aj,i0

i, j∈N^∗

, (24a)

β_i,k,l=β_i,l,k0

i, k, l∈N^∗, i <max{k, l}

, (24b)

b_i,k=b_k,i0, b_1,1=0

i, k∈N^∗, i < k

, (24c)

i<max{k,l}

iβ_i,k,l=k+l

k, l∈N^∗

. (24d)

Because of the quadratic character of the fragmentation terms, the inductive method for the proof of existence devised by Laurençot and Mischler [15] seems difficult to adapt in this case. The method presented in our first appli- cation can however be adapted, provided that the dimension isN=1 and that the following assumptions are made on the coefficients:

Hypothesis 1.11.Assume (24), and suppose that the diffusion coefficients are uniformly bounded above and below (Eq. (13)) and that the initial mass lies inL²(Ω)(Eq. (14)). In place of (7) we assume further that

llim→∞

a_k,l

l =0, lim

l→∞

b_k,l l =0

for fixedk∈N^∗

, (25)

llim→∞sup

k

b_k,l kl βi,k,l

=0

for fixedi∈N^∗

. (26)

We define a solution to (23) along the same lines as in Definition 1.1:

Definition 1.12.A global weak solutionc= {c_i}i1to (23), the boundary condition (1b) and the initial data (1c) is a sequence of functionsc_i: [0,+∞)×Ω→ [0,+∞)such that for eachT >0,

ci∈C

[0, T];L¹(Ω)

, i1, (27)

the four terms on the right-hand side of (23) are inL¹([0, T] ×Ω), sup

t0 Ω

_∞

i=1

ic_i(t, x)

dx

Ω

_∞

i=1

ic⁰_i(x)

dx, (28)

and for eachi1,c_i is a mild solution to thei-th equation in (23), that is,

c_i(t )=e^diA1tc⁰_i +

t

0

e^diA1(t−s)Z_i c(s)

ds, t0,

whereZi[c]represents the right-hand side of (23) andA1,e^diA1t are the same as in Definition 1.1.

We are now able to prove the following theorem:

Theorem 1.13.Under Hypothesis1.11on the coefficients and initial data of the equation, and in dimensionN=1, there exists a global weak solution to Eq.(23)satisfying

c_i∈C

[0, T], L¹(Ω)

∩L^3−ε(Ω_T)

for alli∈N^∗, T >0, ε >0 ,

for which the four terms appearing in the right-hand side of (23)lie inL¹(Ω_T).

Remark 1.14.The method of proof unfortunately does not seem to provide existence in dimensionsN2. Dimension N=2 looks in fact critical as it doesn’t allow a priori a bootstrap in the heat equation with right-hand side inL¹. A possible line of proof could follow [12] in the context of reaction–diffusion equations. In higher dimensionsN3, assuming additionally a detailed balance relation between coagulation and fragmentation, an entropy based duality method as in [9] could be used to define global weakL²solutions (see also [16]).

Our paper is built in the following way: Section 2 is devoted to the proof of Propositions 1.3 and 1.5. Then Sections 3, 4, and 5 are each devoted to one of the three applications. In particular, Theorem 1.9 is proven in Section 4 first in a particular case (with a very short proof), and then in complete generality. Theorem 1.13 is proven in Section 5.

Finally, an Appendix A is devoted to the proof of a lemma of duality due to M. Pierre and D. Schmitt (cf. [17]), which is the key to Proposition 1.3.

2. A new a priori estimate

The solutions given in [15] are constructed by approximating the system (1)–(2) by a truncated system (the proce- dure consists in setting the coagulation and fragmentation coefficients to zero beyond a given finite size, and smoothing the initial data) for which very regular solutions exist. Then, uniform estimates for the solutions of this approximate system are proven. Finally, it is shown that these solutions have a subsequence which converges to a solution to the original system. In the proofs below it must be understood that the bounds are obtained for the truncated system (in a uniform way) and then transferred to the weak solution by a passage to the limit: the fact that this transfer can be done (in the case of the total mass) without replacing the equality by an inequality is the heart of our second application.

We begin with

Proof of Proposition 1.3. Using the fact that

∂_tρ−(Mρ)=0, inf

i∈N^∗{d_i}M(t, x):=

_∞

i=1d_iic_{i ∞}

i=1ici sup

i∈N^∗{d_i},

we can deduce thanks to a lemma of duality [9, Appendix] thatρ∈L²(Ω_T), and more precisely that ρ _L2(Ω_T)

1+sup_i{d_i} infi{d_i}

Tρ(0,·)

L²(Ω),

for allT >0. For the sake of completeness, the lemma is recalled with its proof in Appendix A (Lemma A.2). 2 We now turn to

Proof of Proposition 1.5. ForF_i⁻, it is clear that F_i⁻B_iρ∈L²

[0, T] ×Ω

⊆L¹

[0, T] ×Ω , thanks to Proposition 1.3. ForF_i⁺we use Eq. (16) to write

F_i⁺ ∞ j=1

B_i+jβ_i+j,i i+j

(i+j )ci+jKi

∞ j=1

(i+j )ci+jKiρ, (29)

which is again inL²([0, T] ×Ω), and hence inL¹([0, T] ×Ω).

For the coagulation terms, we have, since eachc_i is less thanρ, Q⁺_i 1

4

i−1

j=1

a_i−j,j

c²_i−j+c²_j

1

2ρ² _i−1

j=1

a_i−j,j

, (30)

which is inL¹([0, T] ×Ω)as the same is true forρ²and the sum consists only of a finite number of terms. Finally, for Q⁻_i we use the fact thatQ⁺_i andF_i⁺are already known to be integrable: Thus, from Eq. (1) integrated over[0, T]×Ω,

Ω

c_i(T , x) dx+

T

0 Ω

Q⁻_i (t, x) dx dt

Ω

c⁰_i(x) dx+

T

0 Ω

Q⁺_i (t, x) dx dt+

T

0 Ω

F_i⁺(t, x) dx dt.

This proves our result. 2

3. First application: A simplified proof of existence of solutions in dimension 1

We begin this section with the following corollary of Proposition 1.5, in the particular case of dimensionN=1.

Lemma 3.1.Assume that the dimensionN=1, and that(3), (13), (14) and(16) (being more general than(7))hold.

Then, for allT 0,i∈N^∗the concentrationsc_i ∈L^∞([0, T] ×Ω)(wherec_i are smooth solutions of a truncated version of (1)–(2), theL^∞norm being independent of the truncation).

Proof. We carry out a bootstrap regularity argument. Thanks to Proposition 1.5, we know that (for alli∈N^∗) (∂_t−d_i)c_i∈L¹

[0, T] ×Ω .

Using for example the results in [8], this implies that for anyδ >0, c_i∈L^3−δ

[0, T] ×Ω i∈N^∗

. (31)

Now, Eq. (31) shows thatQ⁺_i is actually more regular: from (the first inequality in) (30), Q⁺_i ∈L^32−δ2

[0, T] ×Ω

for allδ >0, i∈N^∗. (32)

In addition, we already knew from Eq. (29) that (for alli∈N^∗) F_i⁺∈L²

[0, T] ×Ω

, (33)

[for which we do not need to assume that the space dimensionNis 1]. Consequently, omitting the negative terms (for alli∈N^∗,δ >0), we can findh_i such that

(∂t−di)cihi∈L^32−δ2

[0, T] ×Ω . As thec_i are positive, this implies that

ci∈L^p

[0, T] ×Ω

for allp∈ [1,+∞[, i∈N^∗. Again from (30),

Q⁺_i ∈L^p

[0, T] ×Ω

for allp∈ [1,+∞[, i∈N^∗. From this and (33), we can findhi such that

(∂_t−d_i)c_ih_i∈L²

[0, T] ×Ω ,

which implies in turn thatci∈L^∞([0, T] ×Ω)(for alli∈N^∗). 2

We now have the possibility to give a short proof of Theorem 1.2 in dimension 1 (and under the extra assump- tions (13), (14)). Recall that a proof for any dimension can be found in [15].

Short proof of Theorem 1.2 in 1D assuming additionally (13) and (14). Consider a sequence c^M_i of (regu- lar) solutions to a truncated version of system (1)–(2). Thanks to Lemma 3.1, we know that for each i∈N^∗, sup_M c^M_i L^∞(Ω_T)<+∞. Then (for eachi∈N^∗) there is a subsequence of the (c^M_i )M∈N (which we still denote by(c^M_i )_M∈N), and a functionc_i∈L^∞(Ω_T), such that

c^M_i c^∗ _i weak-∗inL^∞(Ω_T). (34)

Using Proposition 1.5, we also see that (for any fixedi∈N^∗), theL¹(Ω_T)norms ofC⁺_i ^,M,C_i^−,M,F_i^+,M,F_i^−,M(the coagulation and fragmentation terms associated to{c^M_i }) are bounded independently ofM. Using Eq. (1a) and the properties of the heat equation, one sees that for eachi∈N^∗, the sequence{c^M_i }lies in a strongly compact subset of L¹(Ω_T). Hence, by renaming our subsequence again, we may assume that

c^M_i →c_i inL¹(Ω_T)strong , for alli∈N^∗. (35)

In order to prove that{c_i}is indeed a solution to Eq. (1)–(2), let us prove that all termsF_i^+,M,F_i^−,M,C_i^+,M,C_i^−,M converge to the corresponding expressions forc_i, which we denote byF_i⁺,F_i⁻,C_i⁺,C_i⁻, as usual.

1. Positive fragmentation term: for each fixedi, the sum F_i^+,M=

∞ j=1

B_i+jβ_i+j,ic^M_i+j

converges toF_i⁺inL¹(Ω_T)because the tails of the sum converge to 0 uniformly inM(this is due to hypothe- sis (7)):

T

0 Ω

j

B_i+jβ_i+j,i

c_i^M_+j−c_i+jdx dt2

sup

jJ₀

B_i+jβ_i+j,i i+j

ρ+sup

jJ₀

c_i^M_+j−c_i+j

L¹(Ω_T).

2. The negative fragmentation term is just a multiple ofc^M_i , so the convergence inL¹(Ω_T)is given by (35).

3. For each fixedi, the positive coagulation term is a finite sum of terms of the forma_i,jc^M_i c^M_j . Thanks to (34) and (35), this converges toai,jcicjinL¹(ΩT).

4. The negative coagulation term is Q⁻_i ^,M=c^M_i

∞ j=1

a_i,jc_j^M.

Sincec^M_i converges toc_i weak-∗inL^∞(Ω_T), it is enough to prove that_∞

j=1a_i,jc^M_j converges to_∞

j=1a_i,jc_j strongly inL¹(Ω_T). Observing that

T

0 Ω

j

a_i,j

c^M_j −c_jdx dt2

sup

jJ₀

ai,j

j

ρ+ sup

jJ₀

c^M_j −c_j

L¹(ΩT),

we see thanks to (7) and (35) that this convergence indeed holds. 2 4. Second application: Mass conservation

We begin this section with a very short proof of Theorem 1.9 in a particular case in order to show how estimate (15) works. More precisely, we consider the pure coagulation case withai,j=√

ijandBi=0 (no fragmentation), and with initial data satisfying additionally_∞

i=0ilogici(0, x) dx <+∞(which is sightly more stringent than only assuming finite initial mass).

Then, using the weak formulation (4) withϕ_i=log(i)(and remembering that log(1+x)Cst√ x) d

dt

Ω

∞ i=1

ilogicidx=

Ω

∞ i=1

∞ j=1

ij cicj

ilog

1+j

i

+jlog

1+ i j

dx

2

Ω

∞ i=1

∞ j=1

ij c_ic_jdx2

Ω

ρ(t, x)²dx. (36)

As a consequence, we have for allT >0

Ω

∞ i=0

ilogic_i(T , x) dx

Ω

∞ i=0

ilogic_i(0, x) dx+2

T

0 Ω

ρ(t, x)²dx dt,

which ensures the propagation of the moment ^∞_i=0ilogic_i(·, x) dx, and therefore gives a rigorous proof of con- servation of the mass for weak solutions of the system: no gelation occurs.

Our general result is obtained through a refinement of this argument under Hypothesis 1.8. Before giving the proof of Theorem 1.9 we need two technical lemmas, which will substitute the intermediate step in (36).

Lemma 4.1.Let{μ_i}i1and{ν_i}i1be sequences of positive numbers such that{μ_i}is bounded, ∞

i=1

μ_i= +∞ and lim

i→+∞ν_i= +∞.

Then we can find a sequence{ξ_i}i1of nonnegative numbers such that ∞

i=1

ξ_i= +∞,

ξ_iμ_i and ψ_i:=

i

j=1

ξ_jν_i for alli1.

Proof. We may assume thatν_i is nondecreasing, for otherwise we can considerν˜_i:=infji{ν_j}instead ofν_i. Then, in order to findξ_iit is enough to define recursivelyξ₀:=0 and, fori1,

ξ_i:=

μ_i ifμ_i+_i−1

j=0ξ_jν_i, 0 otherwise.

By construction,ξiμi for alli1, and alsoi

j=1ξjνi fori1, as we are assuming{νi}nondecreasing.

To see that{ξ_i}cannot be summable, suppose otherwise that_∞

i=1ξ_i=S <+∞. Take a boundM >0 of{μ_i}, and choose an integerksuch thatν_i S+Mfor allik. Then, by definition,

ξ_i=μ_i for allik,

which implies that{ξ_i}is not summable, as{μ_i}is not, and gives a contradiction. 2

Lemma 4.2.Assume(17). There is a nondecreasing sequence of positive numbers{ψ_i}i1such thatψ_i→ +∞when i→ +∞, and

a_i,j(ψ_i+j−ψ_i)Cj for alli, j1, (37)

for some constantC >0.

In addition, for a given sequence of positive numbersλ_iwithlimi→+∞λ_i= +∞, we can chooseψ_iso thatψ_iλ_i for alli.

Proof. First, we may assume that the functionθ given in Hypothesis 1.8 is nonincreasing on[1,+∞), as we can always takeθ (x)˜ :=sup_yxθ (y)instead.

We choose a sequence of nonnegative numbers{ξ_i}by applying Lemma 4.1 with

μ_i:= 1

(1+i)log(1+i), (38)

ν_i:=min

λ_i, 1 θ (√

i/2),

. (39)

Note that the conditions in Lemma 4.1 are met: the sequence in the right-hand side of (38) is not summable, and the right-hand side of (39) goes to+∞ with i. If we define ψ_i :=_i

j=1ξ_j, then the following is given by Lemma 4.1:

ξ_i 1

(1+i)log(1+i), ψ_i 1 θ (√

i/2), ψ_iλ_i, i1,

i→+∞lim ψ_i= +∞.

These conditions essentially say thatψ_igrows slowlier than log log(i), slowlier thanθ (√

i/2)⁻¹, and slowlier thanλ_i, yet still diverges asi→ +∞.

We can now prove (37) to hold for these{ψ_i}by distinguishing three cases:

1. For anyi, j1, as log(1+k)1/2 for allk1,

ψi+j−ψi=

i+j

k=i+1

ξk2

i+j

k=i+1

1

1+k2 log(i+j+1)−2 log(i+1)2j i .

Then, in casejiwe use the fact thatθ (x)Cθ for some constantCθ>0 and allx >0 and have a_i,j(ψ_i+j−ψ_i)2Cθ(i+j )j

i 4Cθj, forji.

2. Secondly, fori < ji², ψi+j−ψi

2i²

k=i+1

ξk

2i²

k=i+1

1

(k+1)log(k+1) log log

2i²+1

−log log(i+1)log

2 log(√ 3i) log(i+1)

C₁, for some numberC₁>0. Thus,

ai,j(ψi+j−ψi)C1Cθ(i+j )2C1Cθj.

3. Finally, forj > i², ψ_i+j−ψ_iψ_i+j=

i+j

k=1

ξ_k 1

θ (√

(i+j )/2) 1 θ (√

j ),

and asθis nonincreasing on[1,+∞)(we may assume this; see the beginning of this proof), we have for allj > i² a_i,j(ψ_i+j−ψ_i)(i+j )θ (j/ i) 1

θ (√

j )(i+j )θ ( j ) 1

θ (√

j )=i+j2j.

Together, these three cases show (37) for alli, j1. 2

Now we are ready to finish the proof of our result on mass conservation:

Proof of Theorem 1.9. As remarked above (cf. beginning of Section 2), we will prove the estimate (19) for a regular solution to an approximating system, with a constantC(T )that does not depend on the regularisation. Then, passing to the limit, the result is true for a weak solution thus constructed.

We consider a solution to an approximating system on [0,+∞), which we still denote by{ci}i1. Then, by a version of the de la Vallée–Poussin’s Lemma (see, for instance, Proposition 9.1.1 in [4] or also proof of Lemma 7 in [6]), there exists a nondecreasing sequence of positive numbers{λ_i}i1(independent of the regularisation of the initial data) which diverges asi→ +∞, and such that

Ω

∞ i=1

iλic⁰_idx <+∞. (40)

If we defineri:=

Ωic⁰_i, note that this is just the claim that one can findλi as above with

iλiri<+∞.

Taking{ψi}as given by Lemma 4.2, such thatψiλi for alli1, we have thus

Ω

_∞

i=1iψic⁰_i(x) dx <+∞.

Then, as integrating overΩ makes the diffusion term vanish due to the no-flux boundary conditions, we estimate d

dt

Ω

∞ i=1

iψ_ic_idx1 2

Ω

∞ i,j=1

a_i,jc_ic_j

(i+j )ψ_i+j−iψ_i−j ψ_j

dx, (41)

where we used that the contribution of the fragmentation term is nonpositive, as can be seen from (4) withϕi≡iψi, and the fact that

i−1

j=1

β_i,jj ψ_jψ_i

i−1

j=1

β_i,jj=iψ_i,

asψ_iis nondecreasing and (3c) holds. Continuing from (41), by the symmetry of thea_i,j, and using the inequality (37) from Lemma 4.2, we have

d dt

Ω

∞ i=1

iψ_ic_idx

Ω

∞ i,j=1

a_i,jc_ic_ji(ψ_i+j−ψ_i) dxC

Ω

ρ²dx. (42)