The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.

Introduction

The continuous coagulation and multiple-fragmentation equation describes the evolution of the number density of particles of volume at time and reads with The first two terms on the right-hand side of (1) account for the formation and disappearance of particles as a result of coagulation events and the coagulation kernel represents the rate at which particles of volume coalesce with particles of volume . The remaining two terms on the right-hand side of (1) describe the variation of the number density resulting from fragmentation events which might produce more than two daughter particles, and the breakage function is the probability density function for the formation of particles of volume from the particles of volume . Note that it is non-zero only for . The selection function describes the rate at which particles of volume are selected to fragment. The selection function and breakage function are defined in terms of the multiple-fragmentation kernel by the identities The breakage function is assumed here to have the following properties: and The parameter represents the number of fragments obtained from the breakage of particles of volume and is assumed herein to be finite and independent of . The latter is however inessential for the forthcoming analysis; see Remark 2.3. As for the condition (5), it states that the total volume of the fragments resulting from the splitting of a particle of volume equals and thus guarantees that the total volume of the system remains conserved during fragmentation events.

The existence of solutions to coagulation–fragmentation equations has already been the subject of several papers which however are mostly devoted to the case of binary fragmentation, that is, when the fragmentation kernel satisfies the additional symmetry property for all ; see the survey [1] and the references therein. The coagulation–fragmentation equation with multiple fragmentation has received much less attention over the years though it is already considered in the pioneering work [2], where the existence and uniqueness of solutions to (1)–(2) are established for bounded coagulation and fragmentation kernels and . A similar result was obtained later on in [3] by a different approach. The boundedness of was subsequently relaxed in [4] where it is only assumed that grows at most linearly, but still for a bounded coagulation kernel. Handling simultaneously unbounded coagulation and fragmentation kernels turns out to be more delicate and, to our knowledge, is only considered in [5] for coagulation kernels of the form with no growth restriction on and a moderate growth assumption on (depending on ) and in [6] for coagulation kernels satisfying for some sublinear function and a moderate growth assumption on (see also [7] for the existence of solutions for the corresponding discrete model). Still, the fragmentation kernel is required to be bounded near the origin in [6,5] which thus excludes kernels frequently encountered in the literature such as with and  [8].

The purpose of this note is to fill (at least partially) this gap and establish the existence of weak solutions to (1) for simultaneously unbounded coagulation and fragmentation kernels and , the latter being possibly unbounded for small and large volumes. More precisely, we make the following hypotheses on the coagulation kernel , multiple-fragmentation kernel , and selection rate .

is a non-negative measurable function on and is symmetric, i.e. for all ,

for all where for some and constant .

is a non-negative measurable function on such that if . Defining and by (3), we assume that satisfies (4)–(5) and that there are , and two non-negative functions and such that, for each :

we have for and ,

for and any measurable subset of , we have

where denotes the Lebesgue measure of is the indicator function of given by

and we assume in addition that

.

We next introduce the functional setting which will be used in this paper: define the Banach space with norm by together with its positive cone For further use, we also define the norms

The main result of this note is the following existence result:

Suppose that (H1)–(H6) hold and assume that . Then (1)–(2) has a weak solution on in the sense of Definition 1.3. Furthermore, for all .

Before giving some examples of coagulation and fragmentation kernels satisfying (H1)–(H6), we recall the definition of a weak solution to (1)–(2) [9].

Let . A solution of (1)–(2) is a non-negative function such that, for a.e. and all ,

is continuous on ,

the following integrals are finite:

the function satisfies the following weak formulation of (1)–(2):

Coming back to (H1)–(H6), it is clear that coagulation kernels satisfying for some and which are usually used in the mathematical literature satisfy (H1)–(H2); see also [6] for more complex choices. Let us now turn to fragmentation kernels which also fit in the classes considered in Hypothesis 1.1.

Clearly, if we assume that as in [6,3], (H4) and (H5) are satisfied with , and . Now let us take where and ; see [8,10]. Assume that and . Then (4)–(5) are clearly satisfied with and Let us first check (H5). Owing to the constraints on and , we may fix depending only on and such that and , the latter being obviously satisfied when . Given , and a measurable subset of , we deduce from Hölder’s inequality that This shows that (H5) is fulfilled with . As for (H4), for , we write and (H4) is satisfied provided with and if and , and if and , and if and , and , and if and . Therefore, Theorem 1.2 provides the existence of weak solutions to (1)–(2) for unbounded coagulation kernels satisfying (H1)–(H2) and multiple-fragmentation kernels given by (6) with and . Let us however mention that some fragmentation kernels which are bounded at the origin and considered in [6,5] need not satisfy (H4)–(H5). Let us finally point out that, according to [8], the breakage function given by (6) is only physically relevant if , so Theorem 1.2 covers physically realistic breakage functions given by (6) when ranges in .

While the requirement restricting the growth of might be only of a technical nature, the constraint might be more difficult to remove. Indeed, it is well-known that there is an instantaneous loss of matter in the fragmentation equation when and produced by the rapid formation of a large amount of particles with volume zero (dust), a phenomenon referred to as disintegration or shattering [8]. The case thus appears as a borderline case.

Let us finally outline the proof of Theorem 1.2. Since the pioneering work [9], it has been realized that -weak compactness techniques are a suitable way of tackling the problem of existence for coagulation–fragmentation equations with unbounded kernels. This is thus the approach that we use hereafter, the main novelty being the proof of the estimates needed to guarantee the expected weak compactness in . These estimates are derived in Section 2.2 for a sequence of unique global solutions to truncated versions of (1)–(2) constructed in Section 2.1. After establishing weak equicontinuity with respect to time in Section 2.3, we extract a weakly convergent subsequence in and finally show that the limit function obtained from the weakly convergent subsequence is actually a solution to (1)–(2) in Sections 2.4 and 2.5.

Existence

Approximating equations

In order to prove the existence of solutions to (1)–(2), we take the limit of a sequence of approximating equations obtained by replacing the kernel and selection rate by their “cut-off” analogues and  [9], where for . Owing to the boundedness of and for each , we may argue as in [9, Theorem 3.1] or [11] to show that the approximating equation with initial condition has a unique non-negative solution such that for all . In addition, the total volume remains conserved for all , i.e.

From now on, we extend by zero to , i.e. we set for and . Observe that we then have the identity .

Next, we need to establish suitable estimates in order to apply the Dunford–Pettis Theorem [12, Theorem 4.21.2] and then the equicontinuity of the sequence in time to use the Arzelà–Ascoli Theorem [13, Appendix A8.5]. This is the aim of the next two sections.

Weak compactness

Assume that (H1)–(H6) hold and fix . Then we have:

There is (depending on ) such that

For any there exists such that for all

Given there exists such that, for every measurable set of with , and ,

(i) Let and . Integrating (7) with respect to over and using Fubini’s Theorem, we have Since , and are non-negative and satisfies (3), we have Using Fubini’s Theorem and either (H5) (with and ) in the first term of the right-hand side or (H4) (with ) in the second one, we obtain Recalling that for by (9), we readily deduce from (10) that Integrating with respect to time, we end up with Using (9) again we may estimate

(ii) For , set . Then, by (9), for each and for all we have

(iii) Fix . For , and , we define

Consider a measurable subset with . For and , it follows from the non-negativity of and (3) and (7)–(8) that where First, applying Fubini’s Theorem to gives Setting , it follows from (H2) and the above identity that Since and , we infer from the definition of and Lemma 2.1 (i) that Next, applying Fubini’s Theorem to and using (H5) and Lemma 2.1(i) gives Finally, owing to (H4), (9), and Hölder’s inequality, we have Collecting the estimates on , we infer from (11) that there is a such that Integrating with respect to time and taking the supremum over all such that with gives By Gronwall’s inequality (see e.g. [14, p. 310]), we obtain Now, since for , the absolute continuity of the integral guarantees that as which, together with (H5) and (12), implies that Lemma 2.1(iii) is then a straightforward consequence of this property and Lemma 2.1(i). □

Lemma 2.1 and the Dunford–Pettis Theorem imply that, for each , the sequence of functions lies in a weakly relatively compact set of which does not depend on .

Equicontinuity in time

Now we proceed to show the time equicontinuity of the sequence . Though the coagulation terms can be handled as in [6,5,9], we sketch the proof below for the sake of completeness. Let , and , and consider with . Fix such that the constant being defined in Lemma 2.1(i). For each , by Lemma 2.1(i), By (7), (13) and (14), we get

By Fubini’s Theorem, (H2), and Lemma 2.1(i), the first term of the right-hand side of (15) may be estimated as follows: Similarly, for the second term of the right-hand side of (15), it follows from (H2) that For the third term of the right-hand side of (15), we use Fubini’s Theorem, (H4), (H5), and Lemma 2.1(i) to obtain Finally, the fourth term of the right-hand side of (15) is estimated with the help of (H6) and Lemma 2.1(i) and we get Collecting the above estimates and setting the inequality (15) reduces to whenever for some suitably small . The estimate (16) implies the time equicontinuity of the family in . Thus, according to a refined version of the Arzelà–Ascoli Theorem (see [9, Theorem 2.1]), we conclude that there exist a subsequence and a non-negative function such that for all and . In particular, it follows from the non-negativity of and , and (9) and (17), that, for and , Letting implies that and thus .

Passing to the limit

Now we have to show that the limit function obtained in (17) is actually a weak solution to (1)–(2). To this end, we shall use weak continuity and convergence properties of some operators which we define now: for , and , we put and .

We then have the following result:

Let be a bounded sequence in , and such that in as . Then, for each and , we have

The proof of (18) for is the same as that in [6,9] to which we refer. The case is obvious since belongs to by (H6) and (18) follows at once from the weak convergence of in . For , we consider and use (3) and Fubini’s Theorem to compute, for , This can be further written as with

We use (H6) and (4) to observe that, for , This shows that the function belongs to . Since in as , it thus follows that We next infer from (H4) that, for , On the one hand, (21) guarantees that the function belongs to and the weak convergence of to in entails that On the other hand, we deduce from (21) and the boundedness of and in that which is asymptotically small (as ) uniformly with respect to . We thus conclude that Substituting (20) and (22) into (19), we obtain for all . Owing to (23), we may let and conclude that (18) holds true for thanks to the arbitrariness of . The proof of Lemma 2.2 is then complete. □

Existence

Now we are in a position to prove the main result.

Fix , and consider and . Owing to Lemma 2.2, we have for each , Arguing as in Section 2.3, it follows from (H2), (H4)–(H6), and Lemma 2.1(i) that there is a such that, for , and , we have Since the right-hand side of (25) is in , it follows from (24), (25) and the dominated convergence theorem that Since is arbitrary in , Fubini’s Theorem and (26) give It is then straightforward to pass to the limit as in (7)–(8) and conclude that is a solution to (1)–(2) on (since is arbitrary). This completes the proof of Theorem 1.2. □

It is worth pointing out that the assumption (4), , is only used to prove (20) and it is clear from that proof that the assumption is sufficient. Thus, Theorem 1.2 is actually valid under this weaker assumption.