Discrete fragmentation systems in weighted $\ell^1$ spaces

We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $\ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $\ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.


Introduction
There are many diverse situations arising in nature and industrial processes where clusters of particles can merge together (coagulate) to produce larger clusters, and can break apart (fragment) to produce smaller clusters. Particular examples can be found in polymer science, [1,24,25], in the formation of aerosols, [16], and in the powder production industry, [21,23]. It is often appropriate when modelling such processes to regard cluster size as a discrete variable, with a cluster of size n, an n-mer, composed of n identical units (monomers). By scaling the mass, we can assume that each monomer has unit mass and so an n-mer has mass n. The aim is to use the mathematical model to obtain information on how clusters of different sizes evolve. In this paper we restrict our attention to the case when no coagulation occurs, and consequently the evolution of clusters can be described by a linear, infinite system of ordinary differential equations. With the number density of clusters of size n (i.e. mass n) at time t denoted by u n (t), this fragmentation system is given by u ′ n (t) = −a n u n (t) + ∞ j=n+1 a j b n,j u j (t), t > 0; u n (0) =ů n , n = 1, 2, . . . , where a n is the rate at which clusters of size n are lost, b n,j is the rate at which clusters of size n are produced when a larger cluster of size j fragments andů n is the initial density of clusters of size n at time t = 0. Equation (1.1) was first introduced in [25] to deal with the case of binary fragmentation, where it is assumed that each fragmentation event results in the creation of exactly two daughter clusters. As in [7,10,18,19], we consider the more general case, where each fragmentation event can result in the creation of two or more clusters. Since (1.1) is an infinite system, it is convenient to express solutions as time-dependent sequences of the form u(t) := (u n (t)) ∞ n=1 . Throughout this paper we need various assumptions on the fragmentation coefficients a n and b n,j . We list these assumptions here and will refer to them in the sequel when required.
(i) For all n ∈ N, a n ≥ 0. (1.2) (ii) For all n, j ∈ N, b n,j ≥ 0 and b n,j = 0 when n ≥ j. (1. 3) The total mass of daughter clusters resulting from the fragmentation of a j-mer is given by j−1 n=1 nb n,j . In most papers that have dealt with discrete fragmentation systems it is assumed that j−1 n=1 nb n,j ≤ j for all j = 2, 3, . . . , (1.4) i.e. there is no increase in mass at fragmentation events. If there is strict inequality in (1.4), then mass is lost by some other mechanism. However, for most of our results we do not assume that (1.4) holds; this means that mass could even be gained at fragmentation events. We can specify the local mass loss or mass gain with real parameters λ j , j = 2, 3, . . ., such that j−1 n=1 nb n,j = (1 − λ j )j, j = 2, 3, . . . . (1.5) In terms of the densities u n (t), the total mass of all clusters in the system at time t is given by the first moment, M 1 (u(t)), of u(t), where nu n (t). (1.6) A formal calculation establishes that if u is a solution of (1.1), then jλ j a j u j (t). (1.7) The expression in (1.7) gives the rate at which mass may be lost from the system or gained, and also shows that, at least formally, the total mass is conserved when a 1 = 0 and λ j = 0 for all j = 2, 3, . . ., i.e. when a 1 = 0 and j−1 n=1 nb n,j = j for all j = 2, 3, . . . . (1.8) Note that monomers cannot fragment to produce smaller clusters, and hence the case when a 1 > 0 is interpreted as a situation in which monomers are removed from the system. In this paper, the approach we use to investigate (1.1) relies on the theory of semigroups of bounded linear operators, and entails formulating (1.1) as an abstract Cauchy problem (ACP) in an appropriate Banach space. The existence and uniqueness of solutions to the ACP are established via the application of perturbation results for operator semigroups. Of particular relevance is the Kato-Voigt perturbation theorem for substochastic semigroups [5,22] that was first applied to (1.1) in [18], and subsequently in similar semigroup-based investigations into (1.1), such as [8,19]. We use a refined version of this theorem proved by Thieme and Voigt in [20].
In previous studies, including [18,19], the ACP associated with the fragmentation system has been formulated in the space Equipped with the norm n|f n |, f ∈ X [1] , (1.10) X [1] is a Banach space, and f [1] = M 1 (f ) (1.11) if f ∈ X [1] is such that f n ≥ 0, n ∈ N. This means that whenever u : [0, ∞) → X [1] is a non-negative solution of the fragmentation system, the norm, u(t) [1] , gives the total mass at time t. Other Banach spaces, with norms related to higher order moments, have also played a prominent role [8,11], with X [1] being replaced by X [p] , p > 1, where Rather than restricting our investigations to spaces of the type X [p] , we choose to work within the framework of more general weighted ℓ 1 spaces. As we shall demonstrate, this additional flexibility will enable us to establish desirable semigroup properties and results that may not always be possible in an X [p] setting. Therefore, we let w = (w n ) ∞ n=1 be such that w n > 0 for all n ∈ N, and define Equipped with the norm ℓ 1 w is a Banach space, which we refer to as the weighted ℓ 1 space with weight w. Motivated by the terms in (1.1), we introduce the formal expressions and Operator realisations, A (w) and B (w) , of A and B respectively, are defined in ℓ 1 w by and (1. 16) Here, and in the sequel, D(T ) denotes the domain of the designated operator T . Similarly, we shall represent the resolvent, (λI − T ) −1 , of T by R(λ, T ). An ACP version of (1.1), posed in the space ℓ 1 w , can be formulated as Note that this reformulation of (1.1) imposes additional constraints on both the initial data and the sought solutions since we now requireů ∈ ℓ 1 w and also that the solution u(t) ∈ D(A (w) ) ∩ D(B (w) ) for all t > 0. Moreover, as the derivative on the left-hand side of (1.17) is defined in terms of · w , it is customary to look for a solution u ∈ . Such a solution is referred to as a classical solution of (1.17), and has the property that u(t) −ů w → 0 as t → 0 + .
It turns out that often, instead of using the operator A (w) + B (w) on the right-hand side of (1.17), one has to use its closure, which leads to the ACP Yet another option for an operator on the right-hand side is the maximal operator, G (w) max , which is defined by However, the domain of this operator is too large in general to ensure uniqueness of solutions; see Example 4.3 below, and also [6] where a continuous fragmentation equation is studied.
There are a number of benefits to be gained by working in more general weighted ℓ 1 spaces, least of which is the derivation of existence and uniqueness results for (1.1) in ℓ 1 w that reduce to those established in earlier X [p] -based investigations by choosing w n = n p . For example, in Theorem 3.4 we prove that G (w) = A (w) + B (w) is the generator of a substochastic C 0 -semigroup. While this result has already been shown for the specific case w n = n p for p ≥ 1, see [8,18], Theorem 3.4 is formulated for more general weights, and is proved by means of an alternative and novel argument that is based on theory presented in [20]. Our approach also leads to an additional invariance result, which can be used to establish the existence of solutions to the fragmentation system (1.17) for a certain specified class of initial conditions. A further major advantage of working in the more general setting of ℓ 1 w is that it yields results on the analyticity of the related fragmentation semigroups, which do not necessarily hold in the restricted case of w n = n p , p ≥ 1. In particular, in Theorem 5.5 we prove that, for any fragmentation coefficients, we can always find a weight w such that A (w) + B (w) is the generator of an analytic, substochastic C 0 -semigroup on ℓ 1 w . In connection with this, it should be noted that there are no known general results that guarantee the analyticity of the fragmentation semigroup on the space X [1] . Indeed, this provided the motivation for previous investigations into fragmentation ACPs posed in higher moment spaces, which led to a sufficient condition being found in [8] for A (w) + B (w) to generate an analytic semigroup on X [p] for some p > 1. However, simple examples are also given in [8] of fragmentation coefficients where the semigroup is not analytic in X [p] for any p ≥ 1; see Example 5.6.
The importance of establishing the analyticity of the semigroup associated with the fragmentation system is that analytic semigroups have extremely useful properties. For example, if A (w) + B (w) generates an analytic semigroup on ℓ 1 w , then it follows immediately that the ACP (1.17) has a unique classical solution for anyů ∈ ℓ 1 w . In addition, when coagulation is introduced into the system, the analyticity of the semigroup generated by A (w) + B (w) can be used to weaken the assumptions that are required on the cluster coagulation rates to obtain the existence and uniqueness of solutions to the corresponding coagulation-fragmentation system of equations. Such coagulationfragmentation systems will be considered in a subsequent publication.
Once the well-posedness of the fragmentation ACP has been satisfactorily dealt with, the next question to be addressed is that of the long-term behaviour of solutions. Results on the asymptotic behaviour of solutions to (1.17) are given in [7,12,15] for the specific case where the weight is w n = n p for p ≥ 1, n ∈ N. In particular, for massconserving fragmentation processes, where (1.8) holds, it is shown that the solution of (1.17) converges to a state where there are only monomers present if and only if a n > 0 for all n ≥ 2. In Section 6 we continue to work with more general weights and, in the mass-loss case, show that the solution of (1.17) decays to the zero state over time if and only if a n > 0 for all n ∈ N. This mass-loss result can then be used to deduce that the solution, in the mass conserving case, converges to the monomer state if and only if a n > 0 for all n ≥ 2, this result now holding in the general weighted space ℓ 1 w . Regarding the rate at which solutions approach the steady state, the case where mass is conserved and w n = n p for p > 1 is examined in [12,Section 4], and it is shown that solutions decay to the monomer state at an exponential rate, which, however, is not quantified. In Section 6 we obtain results regarding the exponential rate of decay of solutions, both for the mass-conserving and mass-loss cases, by working in a space ℓ 1 w in which A (w) + B (w) generates an analytic semigroup. The approach we use enables us to quantify the exponential decay rate.
In [19], the theory of Sobolev towers is used to investigate a specific example of (1.1) that has been proposed as a model of random bond annihilation. Of particular note is the fact that the resulting analysis provides a rigorous explanation of an apparent nonuniqueness of solutions that emanate from a zero initial condition. We shall establish that an approach involving Sobolev towers can also be used to obtain results on (1.1) for general fragmentation coefficients. By writing (1.1) as an ACP in ℓ 1 w , where w is such that A (w) + B (w) generates an analytic, substochastic C 0 -semigroup on ℓ 1 w , we are able to construct a Sobolev tower and then use this to prove the existence of unique, non-negative solutions of (1.17) for a wider class of non-negative initial conditions than those in ℓ 1 w ; see Theorem 7.2. The paper is structured as follows. In Section 2 we provide some prerequisite results and definitions. Following this, we begin our examination of (1.1) in Section 3, obtaining, in particular, the aforementioned Theorem 3.4, which is then used to draw conclusions on the existence and uniqueness of solutions to (1.17) and (1.18), both in the space X [1] and in more general ℓ 1 w spaces. We consider the pointwise system (1.1) in Section 4 and show that for anyů ∈ ℓ 1 w , a solution of (1.1) can be expressed in terms of the semigroup generated by G (w) = A (w) + B (w) . We then use this result to show that G (w) is a restriction of the maximal operator G max . This is important in investigations into the full coagulation-fragmentation system as it allows the fragmentation terms to be completely described by the operator G (w) . Results on the analyticity of the fragmentation semigroup are presented in Section 5, and then applied both in Section 6, where the asymptotic behaviour of solutions is investigated, and in Section 7, where the theory of Sobolev towers is applied to establish the well-posedness of (1.17) for more general initial conditions.

Preliminaries
We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [9,13]. Let X be a real vector lattice with norm · . The positive cone, X + , of X is the set of non-negative elements in X and, similarly, for a subspace D of X, we denote the set of non-negative elements in D by D + . If X is a vector lattice, then for each f ∈ X the vectors f ± := sup{±f, 0} are well defined and satisfy f + , f − ∈ X + and f = f + − f − . A vector lattice, equipped with a lattice norm · , is said to be a Banach lattice if X is complete under · . Moreover, if the lattice norm satisfies f + g = f + g for all f, g ∈ X + , then X is an AL-space. It can be shown that, when X is an AL-space, there exists a unique, bounded linear functional, φ, that extends · from X + to X; see [9, Theorems 2.64 and 2.65]. We now turn our attention to C 0 -semigroups which are crucial to our investigation into the pure fragmentation system. The notions and results given here can be found in [17]. First we note that if (S(t)) t≥0 is a C 0 -semigroup on a Banach space X, then there exist M ≥ 1 and ω ∈ R such that S(t) ≤ M e ωt for all t ≥ 0, and the growth bound, ω 0 , of (S(t)) t≥0 is defined by Analytic semigroups, see [17,Definition II.4.5], are of particular importance in Section 5. Semigroups of this type have a number of useful properties that make them desirable to work with. For example, if G is the generator of an analytic semigroup, (S(t)) t≥0 , on a Banach space X, then S(t)f ∈ D(G n ) for all t > 0, n ∈ N and f ∈ X, and S(·) is infinitely differentiable.
When dealing with many physical problems, such as the fragmentation system, meaningful solutions must be non-negative, and this requirement has to be taken into account in any semigroup-based investigation. In connection with this, we say that a C 0 -semigroup (S(t)) t≥0 on an ordered Banach space X, such as a Banach lattice, is positive if S(t)f ≥ 0 for all f ∈ X + ; it is called substochastic (resp. stochastic) if, additionally, S(t)f ≤ f (resp. S(t)f = f ) for all f ∈ X + . It follows that if G generates a substochastic semigroup (S(t)) t≥0 , then the associated ACP u ′ (t) = Gu(t), t > 0; u(0) =ů, has a unique, non-negative classical solution, given by u(t) = S(t)ů, for anyů ∈ D(G) + . A result on substochastic semigroups and their generators that we shall exploit is due to Thieme and Voigt,[20,Theorem 2.7]. This result gives sufficient conditions under which the closure of the sum of two operators, such as A (w) + B (w) in (1.17), generates a substochastic semigroup. The existence of an invariant subspace under the resulting semigroup is also established. As we demonstrate below in Proposition 2.4, it is possible to adapt the Thieme-Voigt result to produce a modified version that is ideally suited for applying to the fragmentation system. We first provide some prerequisite results that are used in the proof of this proposition.
Lemma 2.1. Let A be a closable operator in a Banach space X. If G = A is the generator of a C 0 -semigroup on X, then no other extension of A is the generator of a C 0 -semigroup on X.
Proof. Suppose that G = A and H ⊇ A are generators of C 0 -semigroups with growth bounds ω 1 and ω 2 respectively, and assume that H = G. Clearly, H ⊇ G since H is closed. Let λ > max{ω 1 , ω 2 }. Then λ ∈ ρ(G) ∩ ρ(H) and hence λI − G : D(G) → X and λI − H : D(H) → X are both bijective. This is a contradiction since λI − H is a proper extension of λI − G.
The following lemma, which is a special case of [9, Remark 6.6], will also be used. For the convenience of the reader we present a short proof.
Proof. Let f ∈ D(G). Further, let ω 0 be the growth bound of the semigroup generated by G, fix λ > ω 0 and set f 0 : The fact that G generates a positive semigroup implies that R(λ, G) is a positive operator, and therefore g, h ∈ D(G) + . Moreover, which proves the result.
When the fragmentation coefficients satisfy Assumption 1.1 and (1.8), then, as mentioned in the previous section, a formal calculation shows that the total mass is conserved. Consequently, if u is a non-negative solution of the fragmentation system, and it is known that u(t) ∈ X [1] for t ≥ 0, then we would expect u to satisfy [1] for all t ≥ 0.
Clearly this mass-conservation property will hold whenever the solution can be written in terms of a stochastic semigroup on X [1] . To this end, the following proposition will prove useful.
Proposition 2.3. Let (S(t)) t≥0 be a positive C 0 -semigroup on an AL-space, X, with generator G, and let φ be the unique bounded linear extension of the norm · from X + to X.
and the result follows from part (ii).
We now use [20,Theorem 2.7] to obtain the following proposition, which will later be applied to the fragmentation problem.
Also, let φ and φ Z be the linear extensions of · from X + to X and of · Z from Z + to Z respectively. Let A : D(A) → X, B : D(B) → X be operators in X such that D(A) ⊆ D(B). Assume that the following conditions are satisfied.
(a) −A is positive; (b) A generates a positive C 0 -semigroup, (T (t)) t≥0 , on X; (c) the semigroup (T (t)) t≥0 leaves Z invariant and its restriction to Z is a (necessarily positive) C 0 -semigroup on (Z, · Z ), with generator A given by Then there exists a unique substochastic C 0 -semigroup on X which is generated by an extension, G, of A+B. The operator G is the closure of A+B. Moreover, the semigroup Proof. We first show that the conditions of [20, Theorem 2.7] hold. From (ii) and the fact that (Z, · Z ) is an AL-space, it is clear that [20, Assumption 2.5] is satisfied. Also, from (f) and (g) we obtain that

The fragmentation semigroup
In this section, we begin our analysis of the fragmentation system (1.1) by investigating the associated ACP (1.17), which we recall takes the form where A (w) and B (w) are defined in ℓ 1 w by (1.15) and (1.16) respectively. A direct application of Proposition 2.4 will establish that, under appropriate conditions on the weight w, w , we shall refer to (S (w) (t)) t≥0 as the fragmentation semigroup on ℓ 1 w . In the process of proving the existence of the fragmentation semigroup, we shall also obtain explicit subspaces of First we note that ℓ 1 w is an AL-space, with positive cone whenever w = (w n ) ∞ n=1 is a positive sequence. Moreover, in this case the unique bounded linear functional, φ w , that extends · w from (ℓ 1 w ) + to ℓ 1 w is given by We recall also that if we take w n = n for all n ∈ N, then ℓ 1 w = X [1] and · w = · [1] . For this specific case, we shall represent φ w , A (w) and B (w) by M 1 , A 1 and B 1 respectively, and consequently the ACP (1.17) on X [1] will be written as From physical considerations, it is clear that the initial condition,ů, in the ACP (1.17) must necessarily be non-negative, and similarly, if u : [0, ∞) → ℓ 1 w is the corresponding solution, then we require u(t) to be non-negative for all t ≥ 0. Moreover, if we assume (1.4) to hold, or, equivalently, (1.5) with λ j ∈ [0, 1], we expect from (1.7) that mass is either lost or conserved during fragmentation. From (1.6) and the definition of the norm on X [1] , this is equivalent to u(t) [1] ≤ ů [1] for all t ≥ 0, with equality being required in the mass-conserving case, provided that w is such that ℓ 1 w ⊆ X [1] . For convenience, we include the following elementary result which states that the operator A (w) generates a substochastic semigroup on ℓ 1 w for any non-negative weight w.
Lemma 3.1. Let ℓ 1 w and · w be defined by (1.13) and (1.14), respectively, and let (1.2) hold. Then the operator A (w) , defined by (1.15), is the generator of a substochastic For the remainder of this section, the weight, w, will be required to satisfy the following assumption. (i) w n ≥ n for all n ∈ N.
(ii) There exists κ ∈ (0, 1] such that Hence (3.4) is satisfied with κ = 1. In particular, if (1.4) holds, then Assumption 3.2 is automatically satisfied by any weight of the form w n = n p , p ≥ 1.
It is an immediate consequence of Assumption 3.2 that, for any f ∈ D(A (w) ) + , we have Consequently, for all f ∈ D(A (w) ), We now apply Proposition 2.4 to the operators A (w) and B (w) . This involves the construction of a suitable subspace of ℓ 1 w , and to this end we require a sequence (c n ) ∞ n=1 that satisfies c n ≤ c n+1 and a n ≤ c n for all n ∈ N. (3.8) Note that such a sequence can always be found. For example, we can take c n = max{a 1 , . . . , a n } for n = 1, 2, . . . . Let C (w) be the corresponding multiplication operator, defined by and equip D(C (w) ) with the graph norm and hence (ℓ 1 w , · w ) is an AL-space, and the unique linear extension of · w from (ℓ 1 is any other monotone increasing sequence that dominates (a n ) ∞ n=1 , and C is defined analogously to (3.10), then D( C (w) ) ⊆ D(C (w) ).
Proof. We show that the conditions (i), (ii) and (a)-(g) of Proposition 2.4 are all satisfied when A = A (w) , B = B (w) and the AL-spaces (X, · ) and (Z, · Z ) are, respectively, ℓ 1 w and (D(C (w) ), · C (w) ) = (ℓ 1 w , · w ). Clearly, ℓ 1 w is dense in ℓ 1 w and continuously embedded since w n ≤ w n , n ∈ N. It follows that (i) and (ii) both hold.
Condition (a) is obviously satisfied by A (w) , and, for (b), we apply Lemma 3.1 to establish that A (w) generates a substochastic C 0 -semigroup, (T (w) (t)) t≥0 , on ℓ 1 w . It is easy to see that the semigroup (T (w) (t)) t≥0 leaves ℓ 1 w invariant and the generator of the restriction to ℓ 1 w is A ( w) , the part of A (w) in ℓ 1 w ; this shows (c). It is also clear that B (w) is positive. From (3.5) we obtain that, for f ∈ D(A (w) ) + , Hence (d) and (e) hold. Since w n ≥ n, by Assumption 3.2 (i), we have w n = w n +w n c n ≥ n, n ∈ N. Moreover, the monotonicity of (c n ) ∞ n=1 and Assumption 3.2 (ii) imply that This means that Assumption 3.2 also holds for the weight w. Therefore we obtain from (3.7) and (3.13) that D(A ( w) ) ⊆ D(B ( w) ) and φ w ((A ( w) + B ( w) )f ) ≤ 0 for f ∈ D(A ( w) ) + , and so (f) is also satisfied. That (g) holds follows from Thus, the conditions of Proposition 2.4 are all satisfied and therefore G (w) = A (w) + B (w) is the generator of a substochastic C 0 -semigroup, (S (w) (t)) t≥0 , on ℓ 1 w , which also leaves D(C (w) ) = ℓ 1 w invariant.
Finally, assume that (1.8) is satisfied and w n = n for all n ∈ N. Then equality holds in (3.4) with κ = 1 and hence also in (3.5), and so, from Proposition 2.4, the semigroup generated in this case is stochastic.
With the help of Remark 3.5 we obtain the following corollary.
Proof. It follows immediately from Theorem 3.4 that u(t) = S (w) (t)ů is the unique classical solution of (3.15) for allů ∈ D(G (w) ). Moreover, since (S (w) (t)) t≥0 is substochastic, this solution is non-negative ifů ∈ D(G (w) ) + . Now assume that (1.8) holds andů ∈ D(G (w) ) + . Then (S 1 (t)) t≥0 is a stochastic C 0 -semigroup on X [1] . Additionally, since w n ≥ n for all n ∈ N, ℓ 1 w is continuously embedded in X [1] and so, as u(t) is differentiable in ℓ 1 w , u(t) is also differentiable in X [1] and the derivatives must coincide. Moreover, since G (w) is the part of G 1 in ℓ 1 w , we have u(t) ∈ D(G 1 ). Therefore, u(t) = S (w) (t)ů is also a solution of (3.14), and, by uniqueness of solutions, it follows that S (w) (t)ů = S 1 (t)ů for t ≥ 0. Remark 3.5 then establishes that u(t) = S (w) (t)ů is a mass-conserving solution.
Note that even ifů ∈ D(A (w) ), the solution, u(t), of (3.15) need not belong to D(A (w) ) for any t > 0. Hence the existence of a solution of (1.17) is not guaranteed in general; one only has uniqueness of solutions. However, the next theorem shows that under the stronger assumptionů ∈ D(C (w) ) on the initial condition, the ACP (1.17) is well posed.
The next proposition shows that if the sequence (a n ) ∞ n=1 has a certain additional property, then a unique solution of (1.17) exists forů ∈ D(A (w) ).
Proposition 3.8. Let (a n ) ∞ n=1 be an unbounded sequence such that (1.2) holds. Further, define the sequence (c n ) ∞ n=1 by (3.9) and let w = (w n ) ∞ n=1 be such that w n > 0 for all n ∈ N. Then D(C (w) ) = D(A (w) ) if and only if lim inf n→∞ a n c n > 0. (3.16) Proof. Note first that the unboundedness of (a n ) ∞ n=1 implies that c n → ∞ as n → ∞. Since c n ≥ a n for all n ∈ N, we have D(C (w) ) ⊆ D(A (w) ). If (3.16) holds, then there exist γ > 0, N ∈ N such that a n ≥ γc n for all n ≥ N . Let f ∈ D(A (w) ). Then and so D(A (w) ) = D(C (w) ). Now suppose that lim inf n→∞ (a n /c n ) = 0. Then there exists a subsequence, (a n k /c n k ) ∞ k=1 , such that c n k = 0, a n k c n k ≤ 1 k and 1 Let f be such that Then ∞ n=1 a n w n |f n | = ∞ k=1 a n k w n k 1 It follows that f ∈ D(A (w) )\D(C (w) ), showing that D(C (w) ) is a proper subset of D(A (w) ).
Remark 3.9. If (a n ) ∞ n=1 is unbounded and eventually monotone increasing, then (c n ) ∞ n=1 , given by (3.9), satisfies (3.16). Note that, in X [1] , the invariance of D(A (w) ) under the fragmentation semigroup has already been established in [18, Theorem 3.2] for the case when (a n ) ∞ n=1 is monotone increasing. We end this section by obtaining an infinite matrix representation of the fragmentation semigroup (S (w) (t)) t≥0 on ℓ 1 w , which is used in Section 6. Let Assumptions 1.1 and 3.2 be satisfied so that G (w) = A (w) + B (w) is the generator of a substochastic C 0 -semigroup, (S (w) (t)) t≥0 , on ℓ 1 w . For n ∈ N, let e n ∈ ℓ 1 w be given by and let (s m,n (t)) m,n∈N , be the infinite matrix defined by s m,n (t) = (S (w) (t)e n ) m for all m, n ∈ N.
Note that, since (S (w) (t)) t≥0 is positive, s m,n (t) ≥ 0 for all m, n ∈ N. Now, each f ∈ ℓ 1 w can be expressed as f = ∞ n=1 f n e n , where the infinite series is convergent in ℓ 1 w . Hence and therefore (S (w) (t)) t≥0 can be represented by the matrix (s m,n (t)) m,n∈N . To determine s m,n (t) more explicitly, fix n ∈ N and let (u 1 (t), . . . , u n (t)) be the unique solution of the n-dimensional system For m = n, the differential equation in (3.19) reduces to u ′ n (t) = −a n u n (t), which implies that s n,n (t) = u n (t) = e −ant . Since n was arbitrary, it follows that, for all t ≥ 0, where 0 is an infinite column vector consisting entirely of zeros, S (w) (12) (t) is a non-negative infinite row vector and S (w) (22) (t) is an infinite-dimensional, non-negative, upper triangular matrix. We note that, in the particular case when ℓ 1 w = X [1] and mass is conserved, Banasiak obtains the infinite matrix representation (3.22) for the semigroup (S 1 (t)) t≥0 in [7, Equation (10) and Lemma 1]. In [7], an explicit expression is also found for s m,n (t), m < n, but we omit this here since it is not required for the results that follow. As observed in [7, pp. 363], it follows from (3.22) that, for all N ∈ N, we have S(t)f ∈ span{e 1 , e 2 , . . . , e N } for all f ∈ span{e 1 , e 2 , . . . , e N }. Also note that the functions s m,n are independent of the weight w, which implies that, whenever w is another weight satisfying Assumption 3.2, S (w) (t) and S ( w) (t) coincide on ℓ 1 w ∩ ℓ 1 w .

The pointwise fragmentation problem and the fragmentation generator
We established in Theorem 3.7 that if Assumptions 1.1 and 3.2 are satisfied, then u(t) = S (w) (t)ů is the unique, non-negative classical solution of the fragmentation ACP (1.17) for allů ∈ D(C (w) ) + . Moreover, when (1.8) holds, then this solution is mass conserving. Clearly, u(t) = S (w) (t)ů will also satisfy the fragmentation system (1.1) in a pointwise manner whenů ∈ D(C (w) ) + . However, at this stage we do not know in what sense, if any, the semigroup (S (w) (t)) t≥0 provides a non-negative solution for a generalů ∈ (ℓ 1 w ) + . In this section we show that a non-negative solution of the pointwise system (1.1) can be determined for any given initial condition in (ℓ 1 w ) + by using the semigroup (S (w) (t)) t≥0 . As before, we require Assumptions 1.1 and 3.2 to hold, and we define a sequence (c n ) ∞ n=1 by (3.9), with the associated multiplication operator C (w) given by (3.10). Then a n ≤ c n for all n ∈ N and it follows that D(C (w) ) ⊆ D(A (w) ). From Proposition 3.4, D(C (w) ) is invariant under the substochastic semigroup (S (w) (t)) t≥0 generated by G (w) = A (w) + B (w) . Consequently, u(t) = S (w) (t)ů is the unique, non-negative classical solution of (1.17) for eachů ∈ D(C (w) ) + , and therefore u n (t) −ů n = −a n  Proof. Letů ∈ (ℓ 1 w ) + and, for N ∈ N, define the operator P N : ℓ 1 w → ℓ 1 w by Then P Nů ∈ D(C (w) ) + for all N ∈ N, and so, on setting u (N ) (t) = S (w) (t)P Nů , we have u (N ) n (t) = P Nůn − a n for n = 1, 2, . . . , N . Clearly, P Nů →ů in ℓ 1 w as N → ∞, and so, by the continuity of S (w) (t), it follows that u (N ) n (t) → u n (t) as N → ∞ for all n ∈ N and t ≥ 0. Moreover, if N 2 ≥ N 1 then u (N2) (t) − u (N1) (t) ≥ 0 for all t ≥ 0, since (S (w) (t)) t≥0 is linear and positive. Similarly, u(t) − u (N ) (t) ≥ 0 for all N ∈ N and t ≥ 0. Hence (u (N ) (t)) ∞ N =1 is monotone increasing and bounded above by u(t), and therefore, for each fixed n ∈ N, (u (N ) n (t)) ∞ N =1 is monotone increasing and bounded above by u n (t). On allowing N → ∞ in (4.2), and using the monotone convergence theorem, we obtain u n (t) =ů n − a n t 0 u n (s) ds + lim From this, we deduce that Thus, for allů ∈ (ℓ 1 w ) + , It follows that (S (w) (t)ů) n is absolutely continuous with respect to t for each n = 1, 2, . . . and so for allů ∈ (ℓ 1 w ) + and almost every t ≥ 0. Whenů is a general, and therefore not necessarily non-negative, sequence in ℓ 1 w , we can expressů =ů + −ů − ∈ ℓ 1 w . It then follows immediately from the first part of the proof that u(t) = S (w) (t)ů also satisfies (1.1) for almost all t ≥ 0.
The last statement of the theorem follows immediately from the positivity of the semigroup (S (w) (t)) t≥0 .
Note that, in general, solutions of (1.1) are not unique; see the discussion in Example 4.3 below.
We now turn our attention to obtaining a simple representation of the generator G (w) . Although we know that G (w) coincides with A (w) + B (w) on D(A (w) ), and also that u(t) = S (w) (t)ů is the unique classical solution of (3.15) forů ∈ D(G (w) ), we have yet to ascertain an explicit expression that describes the action of G (w) on D(G (w) ). This matter is resolved by the following theorem, which shows that G (w) is a restriction of the maximal operator, G (w) max , defined in (1.19). In the specific case of X [p] , the result has been obtained from [9,Theorem 6.20], which uses extension techniques first introduced by Arlotti in [4], and which is applied in [8, Theorem 2.1]. We present an alternative proof, which avoids the use of such extensions. Proof. It follows from Lemma 2.2 and its proof that, for every g ∈ D(G (w) ), there exist g 1 , g 2 ∈ D(G (w) ) + such that g = g 1 − g 2 and f j := (I − G (w) )g j ∈ (ℓ 1 w ) + for j = 1, 2. This and the linearity of G (w) allow us to assume that g ∈ D(G (w) ) + such that f := (I − G (w) )g ∈ (ℓ 1 w ) + . Defining u(t) = S (w) (t)f , we have from (4.3) that By Tonelli's theorem, we have ∞ 0 t 0 e −t a n u n (s) ds dt = ∞ 0 ∞ s e −t a n u n (s) dt ds = a n ∞ 0 e −s u n (s) ds = a n R(1, G (w) )f n .
Using Tonelli's theorem and the monotone convergence theorem we obtain Thus and (4.5) follows.
We note that the formula (4.5) is independent of the weight w = (w n ) ∞ n=1 . Being able to express the action of G (w) in this way is important when investigating the full coagulation-fragmentation system, as it enables the fragmentation terms to be described by means of an explicit formula for the operator G (w) . We shall return to this in a subsequent paper.
u n (0) =ů n , n = 1, 2, . . . , (4.6) which coincides with (1.1) if one sets a n = n − 1, b n,j = 2 j − 1 , n, j ∈ N, j > n. (4.7) The system (4.6) models random scission; see, e.g. [25, equation (49)] and [14, equation (10)]. It is easily seen that (1.8) is satisfied, and hence, mass is conserved. The example (4.6) is closely related to the example that is studied in [19, §3] and which models random bond annihilation. More precisely, if we denote the operators for the example from [19] by A (w) , B (w) , G (w) etc., then and hence S (w) (t) = e t S (w) (t), t ≥ 0. For the particular case when w n = n, n ∈ N, we have similar relations for the operators A 1 , A 1 etc. It follows from [19, Lemma 3.6] that every λ > 0 is an eigenvalue of the maximal operator G 1,max (i.e. the operator G The existence of positive eigenvalues of G 1,max implies that G 1,max is a proper extension of G 1 . Note that the domain of G 1 is determined explicitly in [19,Theorem 3.7], from which we obtain that Using the eigenvectors g (λ) from (4.8) we can define the function which is a solution of the ACP . On the other hand, since the semigroup (S 1 (t)) t≥0 is analytic by [19,Theorem 3.4], the function u(t) = S 1 (t)g (λ) , t ≥ 0, is also a solution of (4.10) and is distinct from u (λ) . This shows that, in general, one does not have uniqueness of solutions of the ACP, (4.10), corresponding to the maximal operator, G 1,max , and hence, also solutions of (1.1) are not unique.
More generally, a specific characterisation of D(G (w) ) is given by [9, Theorem 6.20], but this does not lead to an explicit description, such as that obtained in Example 4.3.

Analyticity of the fragmentation semigroup
In Section 3 we established that Assumptions 1.1 and 3.2 are sufficient conditions for G (w) = A (w) + B (w) to be the generator of a substochastic C 0 -semigroup, (S (w) (t)) t≥0 , on ℓ 1 w . This enabled us to obtain results on the existence and uniqueness of solutions to (1.17). We now investigate the analyticity of (S (w) (t)) t≥0 and prove that, given any fragmentation coefficients, it is always possible to construct a weight, w, such that A (w) + B (w) is the generator of an analytic, substochastic C 0 -semigroup on ℓ 1 w . This particular result, which is one of the main motivations for carrying out an analysis of the fragmentation system in general weighted ℓ 1 spaces, requires a stronger assumption on the weight w. Note that when dealing with analytic semigroups, we use complex versions of the spaces ℓ 1 w .
(i) w n ≥ n for all n ∈ N.
Note that Assumption 5.1 is obtained from Assumption 3.2 by simply replacing κ ∈ (0, 1] with κ ∈ (0, 1). By removing the possibility of κ = 1, we can obtain the following improved version of Theorem 3.4.
(ii) If there exists λ 0 > 0 such that (1.5) holds with λ j ≥ λ 0 for all j ≥ 2 (which corresponds to a 'uniform' mass loss case), then Assumption 5.1 immediately holds with w n = n for all n ∈ N, and κ = 1 − λ 0 .
The following lemma gives sufficient conditions under which Assumption 5.1 holds.
This leads to the main result of this section.
Theorem 5.5. For any given fragmentation coefficients for which Assumption 1.1 holds we can always find a weight, w = (w n ) ∞ n=1 , such that A (w) + B (w) is the generator of an analytic, substochastic C 0 -semigroup on ℓ 1 w . If, in addition, (1.4) holds, we can choose w n = r n with arbitrary r > 2 and κ = 2/r so that (5.1) holds.
Proof. For the first statement note that we can choose w n ≥ n iteratively so that (5.1) is satisfied. The claim then follows from Theorem 5.2. Now assume that (1.4) holds. Let r > 2, w n = r n for n ∈ N, and δ = 2/r, which satisfies δ < 1. Then w n ≥ n and δ w n+1 n + 1 = 2 r · r n+1 n + 1 = 2r n n + 1 ≥ 2r n n + n = r n n = w n n , which shows that (5.2) is satisfied. Hence Lemma 5.4 implies that Assumption 5.1 is fulfilled.
As mentioned earlier, analytic semigroups have a number of desirable properties, and Theorem 5.5 will play an important role when we investigate the full coagulationfragmentation system in a subsequent paper. In particular, Theorem 5.5 will enable us to relax the usual assumptions that are imposed on the coagulation rates in order to obtain the existence and uniqueness of solutions to the full coagulation-fragmentation system.
It should be noted that a condition that is equivalent to Assumption 5.1 has previously been used as a condition for analyticity in the mass-conserving case by Banasiak; see [8,Theorem 2.1]. However, the choice of weights in [8] is restricted to w n = n p , p > 1, and Assumption 5.1 need not be satisfied for these weights for any p > 1 as the following example shows.
Example 5.6. Consider the mass-conserving case where a cluster of mass n breaks into two clusters, with respective masses 1 and n − 1. The corresponding fragmentation coefficients take the form For the choice a 0 = 0; a n = n, n ≥ 2; w n = n p , n ∈ N; p ≥ 1, it is proved in [7, Theorem 3] (for p = 1) and [8,Theorem A.3] (for p > 1) that the semigroup generated by G (w) is not analytic. On the other hand, Theorem 5.5 guarantees the existence of exponentially growing weights w n such that G (w) = A (w) + B (w) generates an analytic semigroup. It is easy to show that for this particular example one can also choose powers of 2, namely, w 1 = 1 and w n = 2 n for n ≥ 2, in which case κ = 5/8.

Asymptotic behaviour of solutions
There have been several earlier investigations into the long-term behaviour of solutions to the mass-conserving fragmentation system (1.1), when (1.8) holds. In particular, the case of mass-conserving binary fragmentation is dealt with in [15], where it is shown that, under suitable assumptions, the unique solution emanating fromů must converge in the space X [1] to the expected steady-state solution M 1 (ů)e 1 , where M 1 (ů) and e 1 are given by (1.6) and (3.18) respectively. This was followed by [7,12] where, once again, the expected long-term steady-state behaviour is established, but now for the massconserving multiple-fragmentation system. More specifically, in [7], a semigroup-based approach is used to prove that, for anyů ∈ X [1] , lim t→∞ S 1 (t)ů − M (ů)e 1 [1] = 0 if and only if a n > 0 for all n = 2, 3, . . . .
That the corresponding result is also valid in the higher moment spaces X [p] , p > 1, is established in [12], and, under additional assumptions on the fragmentation coefficients, it is shown in [12,Theorem 4.3] that there exist constants L > 0 and α > 0 such that the fragmentation semigroup (S p (t)) t≥0 on X [p] , p > 1, satisfies for allů ∈ X [p] . It follows from [3], that the fragmentation semigroup (S p (t)) t≥0 has the asynchronous exponential growth (AEG) property (with λ * = 0 in [3, equation (3)], i.e. with trivial growth). The assumptions required in [12] to prove that (6.1) holds in some X [p] space are somewhat technical and not straightforward to check. Moreover no information on the size of the constant α, and hence the exponential rate of decay to the steady state, is provided. Our aim in this section is to address these issues.
Working within the framework of more general weighted ℓ 1 spaces, we study the longterm dynamics of solutions in both the mass-conserving and mass-loss cases. When mass is conserved, we establish simpler conditions under which the fragmentation semigroup (S (w) (t)) t≥0 satisfies an inequality of the form (6.1) on some space ℓ 1 w , and also quantify α.
On the other hand if we choose α > a 0 , then there exists N ∈ N such that a N < α, in which case (S (w) (t)e N ) N = e −aN t > e −αt for t > 0, and so e αt S (w) (t)e N w ≥ e αt w N S (w) (t)e N N > e αt w N e −αt = w N .
Hence (6.4) cannot hold for any α > a 0 . Remark 6.2. When the assumptions of Theorem 6.1 are satisfied and a n > 0 for all n ∈ N, then (6.2) shows that the only equilibrium solution of (1.17) is u(t) ≡ 0, and this equilibrium is a global attractor for the system. On the other hand, if a n = 0 for at least one n ∈ N, then u(t) ≡ 0 is not a global attractor.
We now examine the mass-conserving case and assume that (1.8) holds. Note that, in this mass-conserving case, the fragmentation semigroup (S 1 (t)) t≥0 is stochastic on the space X [1] . Our aim is to establish an ℓ 1 w version of the results obtained in [7,12,15]. To this end, we recall the matrix representation of S (w) (t) given by (3.22), and also define a sequence space Y (w) , and its norm · Y (w) , by respectively. Clearly, Y (w) is a weighted ℓ 1 space, and can be identified with ℓ 1 w , where w n = w n+1 for n ∈ N. Moreover, we define the embedding operator J : Y (w) → ℓ 1 w by Lemma 6.3. Let α ≥ 0 and f ∈ ℓ 1 w be fixed, and definef := (f n ) ∞ n=2 . If Assumptions 1.1, 3.2 and (1.8) hold, then for all t ≥ 0.
Proof. It follows from (3.22) that From this we deduce that and so , which is the first inequality in (6.5).
We are now in a position to prove the main theorem of this section. The first part confirms that S (w) (t)ů → M 1 (ů)e 1 in ℓ 1 w as t → ∞, for allů ∈ ℓ 1 w , provided that Assumption 3.2 holds and the fragmentation rates, a n , are positive for all n ≥ 2. In the second part, which deals with quantifying the rate of convergence to equilibrium, the fragmentation coefficients are assumed additionally to be bounded below by a positive constant, and Assumption 3.2 is strengthened to Assumption 5.1. In this case, the decay to zero of S (w) (t)ů − M 1 (ů)e 1 w is shown to occur at an exponential rate, defined explicitly in terms of the rate coefficients and the constant κ ∈ (0, 1) in Assumption 5.1. for allů ∈ ℓ 1 w if and only if a n > 0 for all n ≥ 2.
Proof. Removing the equation for u 1 from (1.1) leads to a reduced fragmentation system that can be formulated as an ACP in Y (w) = ℓ 1 w , where, as before, w n = w n+1 for all n ∈ N. The fragmentation coefficients, ( a n ) ∞ n=1 and ( b n,j ) n,j∈N:n<j , associated with the reduced system are given by a n = a n+1 and b n,j = b n+1,j+1 . Clearly, a n ≥ 0 and b n,j ≥ 0 for all n, j ∈ N and b n,j = 0 if n ≥ j, and w n = w n+1 ≥ n + 1 > n for all n ∈ N. Moreover, for j = 2, 3, . . ., Hence Assumptions 1.1 and 3.2 are satisfied by w, a n and b n,j , and it follows from Theorem 3.4 and (3.22) that associated with the reduced system is a substochastic C 0 -semigroup on Y (w) , which can be represented by the infinite matrix An inspection of (3.19), together with (3.21), shows that s m,n (t) = s m+1,n+1 (t) for all n ∈ N, m = 1, . . . , n − 1, t ≥ 0, and therefore the substochastic semigroup on Y (w) is given by (S if and only if a n > 0 for all n ≥ 2, and the result is then an immediate consequence of Lemma 6.3. (ii) The calculations above show that, when Assumption 3.2 holds for w and the coefficients (b n,j ), it is also satisfied by w and ( b n,j ) with exactly the same value of κ. Therefore, from Theorem 6.1, S (w) (22) (t) ≤ e −(1−κ) a0t , and (6.9) follows immediately from Lemma 6.3. Moreover, if a 0 > 0 and α ∈ [0, (1 − κ) a 0 ), then we obtain (6.10).
Remark 6.5. When the assumptions of Theorem 6.4 are satisfied, then it follows from (3.22) that u M = M e 1 is an equilibrium solution of the mass-conserving fragmentation system for all M ∈ R. In addition, the basin of attraction for u M is given by {ů ∈ ℓ 1 w : M 1 (ů) = M } provided that the assumptions of Theorem 6.4 hold and a n > 0 for all n ≥ 2. On the other hand, if a N = 0 for some N ≥ 2, then M e N is also an equilibrium solution for every M ∈ R.

Sobolev towers
In this section we use a Sobolev tower construction to obtain existence and uniqueness results relating to the pure fragmentation system for a larger class of initial conditions. Sobolev towers appear to have been first applied to the discrete fragmentation system (1.1) in [19], where the authors examine a specific example and use Sobolev towers to explain an apparent non-uniqueness of solutions. As we demonstrate below, the theory of Sobolev towers is applicable to more general fragmentation systems and, in the following, the only restrictions that are imposed are that the fragmentation coefficients satisfy Assumption 1.1, and also that a weight, w = (w n ) ∞ n=1 , has been chosen so that Assumption 5.1 holds. These restrictions imply that G (w) = A (w) +B (w) is the generator of an analytic, substochastic C 0 -semigroup, (S (w) (t)) t≥0 , on ℓ 1 w . Let ω 0 be the growth bound of (S (w) (t)) t≥0 . Choosing µ > ω 0 , we rescale (S (w) (t)) t≥0 to obtain an analytic semigroup, (S (w) (t)) t≥0 = (e −µt S (w) (t)) t≥0 , with a strictly negative growth bound. The generator of (S (w) (t)) t≥0 is G (w) = G (w) − µI. We set X (w) 0 = ℓ 1 w , · 0 := · w , S  f n = (G (w) ) n f w , f ∈ D (G (w) ) n , n ∈ N.
For each n ∈ N, X (w) n is referred to as the Sobolev space of order n associated with the semigroup (S (w) (t)) t≥0 . We also define the operator G for each n ∈ N. Sobolev spaces of negative order, −n, n ∈ N, are defined recursively by where (X, · ) denotes the completion of the normed vector space (X, · ). Operators G  Recalling that we also assume that w n ≥ n for all n ∈ N, we have that ℓ 1 w is continuously embedded in X [1] , and from this we deduce that if u(t) is differentiable with respect to the norm on ℓ 1 w then it is also differentiable with respect to the norm on X [1] , and the derivatives coincide. Since A 1 + B 1 is an extension of G (w) = A (w) + B (w) , we conclude that u(t) = S (w) −n (t)ů also satisfies (3.2).