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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 484391, 9 pages
Global Analysis of a Discrete Nonlocal and Nonautonomous Fragmentation Dynamics Occurring in a Moving Process
Department of Mathematical Sciences, North-West University, Mafikeng 2735, South Africa
Received 12 September 2013; Accepted 28 September 2013
Academic Editor: Abdon Atangana
Copyright © 2013 E. F. Doungmo Goufo and S. C. Oukouomi Noutchie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We use a double approximation technique to show existence result for a nonlocal and nonautonomous fragmentation dynamics occurring in a moving process. We consider the case where sizes of clusters are discrete and fragmentation rate is time, position, and size dependent. Our system involving transport and nonautonomous fragmentation processes, where in addition, new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of forward propagators associated with evolution semigroup theory and perturbation theory. The full generator is considered as a perturbation of the pure nonautonomous fragmentation operator. We can therefore make use of the truncation technique (McLaughlin et al., 1997), the resolvent approximation (Yosida, 1980), Duhamel formula (John, 1982), and Dyson-Phillips series (Phillips, 1953) to establish the existence of a solution for a discrete nonlocal and nonautonomous fragmentation process in a moving medium, hereby, bringing a contribution that may lead to the proof of uniqueness of strong solutions to this type of transport and nonautonomous fragmentation problem which remains unsolved.
1. Introduction and Useful Definitions
Fragmentation models have attracted considerable attention lately as they can be found in many important areas of science and engineering. Examples range from the splitting of phytoplankton clusters, astrophysics, rock crushing, colloidal chemistry, and polymer science to depolymerization. The dynamical behavior of a nonautonomous system of phytoplankton clusters, for example, which are undergoing breakup to produce smaller particles in a moving medium can be derived by balancing loss and gain of clusters of size (with position ) over a short period of time and is given by the following differential equation as presented in : where is the particle mass distribution function with respect to the mass at the position and time , ( is the mass distribution at some fixed time , is the distribution of particle masses and position , spawned by the fragmentation of a particle of mass at time , , and is the evolutionary time-dependent fragmentation rate, that is, the rate at which mass particles at position break up at a time . The velocity of the transport is supposed to be a known quantity, depending on the size of aggregates and their position .
The combination of fragmentation equations and transport mechanisms have been successfully utilized to model a wide range of natural processes. Examples in chemical engineering include polymer breakup and solid drugs degradation in fluids. In aquaculture, such models are used to describe phytoplankton dynamics under the kinetic constraints of the flow. The mathematical investigation of fragmentation models presents several challenges both from the theoretical and numerical point of view. In  the authors investigated the existence of global solutions to continuous nonlocal convection-fragmentation equations in spaces of distributions with finite higher moments. But till now, models with time dependent coefficients (nonautonomous) remain barely touched. Moreover, models of transport and nonautonomous fragmentation process have not yet been studied in the same work and there are still only few papers devoted to their analysis (well-posedness, conservativeness, honesty,…) separately or in the same model. In , the authors used techniques of truncation to prove existence, uniqueness, and mass conservation for a pure nonautonomous fragmentation model under certain conditions on initial data and the associated truncated system. The authors in  used evolution semigroups approach which allows them to build on the substochastic semigroup theory and obtain an equivalent result as in . In the analysis of the book by Kato  and later improved by Da Prato and Grisvard , it is generally assumed that the generators and involved in the perturbation are of class . This condition is modified in  where the authors used semigroup perturbation and renormalization approach to show that the closure of the involved operators is an antigenerator. However, in many applications of forward propagator (evolution semigroup) theory to evolution equations like transport equations used in this paper or population equations [8, 9], perturbation method remains essential no matter which generator is the perturbed or the perturbing operator.
As in , we focus on the case where after clusters fragmentation, new originating groups have different centers distributed according to a given probabilistic law . This is the probability density that after a break up of an -aggregate (with the center at ), the new formed -group will be located at . Therefore and the system (1) becomes Since a group of size cannot split to form a group of size , we require at any time and position for all . We also set meaning that a cluster of size one cannot split and the sum of all individuals obtained by fragmentation at a position of an -group is equal to all the time . The second term on the right-hand side of (3) describes the increase in the number of particles of size due to fission of larger particles (the gain due to the fragmentation). The third term describes the reduction in the number of particles of size due to the fission of groups of the same size (the loss due to the fragmentation). The space variable is supposed to vary in the whole of .
2. Approximation and Analysis of the Fragmentation Operator
Since is the density of -groups at the position and time and that total mass is expected to be a conserved quantity, the most appropriate Banach space to work in is the space where is equipped with the weighed counting measure with weight and is the Lebesgue measure in . Note that elements’ norm of represents the total mass (or total number of individuals) of the system.
Now we recast (3) as the nonautonomous abstract Cauchy problem in : or in the compact form where is the vector , the mass distribution vector at the fixed time and position , and the nonautonomous fragmentation operator defined by where is seen as the pointwise operation defined on the set of measurable functions. is defined by and represents the realization of on the domain with given by (11). The transport matrix is given by with To investigate and analyze the abstract Cauchy Problem (8) and show existence for this system, we will need a two parameter family.
Definition 1 (evolution system  or propagator ). A two parameter family of bounded linear operators , , on a Banach space is called forward propagator or evolution system if the following conditions are respected:(i) for ;(ii) for ;(iii) is strongly continuous for .
Recall that  the forward propagator , can be associated with the so-called evolution semigroup defined in ; that is, if for any fixed , the operator generates a forward propagator , , then this propagator defines a semigroup given by the relation where is the characteristic function of , , and . In the following pages, when we say is the generators of semigroups in , we mean that generates a propagator which defines a semigroup in satisfying the relation (14).
We introduce, for any given , the projection operator defined for a function as The space is therefore a closed subspace of on which the projection operator acts. Let us associate with the fragmentation model the following truncated version: where is represented by (11). We set , then can be decomposed as where the loss and the gain fragmentation operators and are given by where and are expressed as Thus, for all , and ,with We assume that for fixed in , there are two constants and such that where is a real-valued function which can depend on and but is independent of the state variable . This obviously implies that for any there exists a positive such that Moreover the sequence ( fixed in ) is bounded in the following way:
Lemma 2. For fixed in , and , there is a positive uniformly continuous semigroup of contractions on , say generated by operator such that is conservative on and given by Moreover, for any , ,
Proof. Let us fix in . The operator is bounded by (25). Changing the order of summation by the Tonelli’s theorem, for every ,
where we have used (2) and (5), respectively. Then is also bounded. Hence generates a uniformly continuous semigroup. We denote this semigroup by . Clearly, generates a positive semigroup of contractions and is a positive operator. Moreover, the above calculations also imply that and
for all with
Thus the assumptions of Kato’s theorem in -space [12, Corollary 5.17] hold. We essentially note that for each fixed the operator becomes independent of time [3, Lemma 2.1] and Kato’s theorem is immediately applicable. Therefore there is an extension of which generates a substochastic semigroup. Because is bounded in , this substochastic semigroup is conservative and it follows that , where is the closure of . Since generates a uniformly (and hence strongly) continuous semigroup, is a closed operator. Therefore we have that ; consequently, the uniformly continuous semigroup is a positive strongly continuous semigroup of contractions and furthermore is honest.
The proof of (27) is clear since we can use the usual power series definition to define . By induction, for , from which the exponential formula yields (27).
To prove (28), we have that on since for , is given by (21) and for . Moreover, it is obvious that ; hence we have also Next, by we have if we assume, by induction, that , then Now using (27) and the the fact that is a bounded operator, the semigroup generated by is expressed by which concludes the lemma.
Next we assume that satisfies the Lipschitz condition where together with ; for all we state the following lemma.
Lemma 3. The function is continuous in the uniform operator topology for each fixed in .
Proof. Using Fubini’s theorem and assumption (36) yields and the result follows.
Making use of (14) and Lemma 2, there is a forward propagator, say defined in which is associated with the evolution semigroup , . The propagator shares certain properties with the family of semigroups , , as stated in the following theorem and proved in [3, Theorem 4.1].
Theorem 4 (see ). For each , the forward propagator generated by the family of infinitesimal generators have the following properties:(1) is positive;(2), for all ;(3), for all , ;(4), ;(5), .
Theorem 5. The truncated problem (18) has a unique, strongly continuously differentiable, positive, mass-conserving solution for all initial data . The solution is given by .
Remark 6. According to [11, Definition 2.1, page 130, Remark 3.2, page 138], for each fixed , the family of infinitesimal generators of -semigroups on is stable with stability constants 1 and 0. Moreover we have from Theorem 4
3. Cauchy Problem for the Transport Model in
We consider the Cauchy problem for the transport equation in the space .
We consider the function defined by and recall that in (13) is the expression appearing on the right-hand side of (39). Then We assume that (H1) is divergence-free;(H2) globally Lipschitz continuous; then and (40) becomes We define in the operators as
Remark 7. With the assumptions (H1)-(H2), it is proven  that is the generator of a strongly continuous stochastic semigroup, say and . Then and due to Hille Yosida’s characterization, is a closed and densely defined operator satisfying for the resolvent set and for some constant and all . Furthermore, (i)for any , (ii) are bounded operators and for any ,
4. Perturbed Approximated Problem
Let ; Yosida  was the first to use the bounded operators to approximate the unbounded operator , for which we can define semigroups via the exponential formula We exploit this idea to analyze the following approximated problem associated with (3):
Lemma 8. Let each be fixed in and any . Under the assumptions (5), (25), (26), (36), (H1) and (H2), the operator , appearing in approximated Cauchy problem (46) is a stable generator, with the stability constants and , of a forward propagator associated with an evolution semigroup , positive, conserving the norm and given by
with defined in Lemma 2.
Furthermore the problem (46) has a unique, strongly continuously differentiable, positive, mass-conserving solution for all initial data . The solution is given by .
Proof. Let us fix and . The fact that is the generator of a forward propagator comes from Lemma 2, the Bounded perturbation theorem, and the remark (14). By Remark 6 we have . If , it is obvious that (bounded perturbation) and the resolvent satisfies
Henceforth, for any finite sequence , , we have
The development of the right-hand side of (50) yields a series with the general term in the form
where . If ; then using the stability of the family , we estimate (51) by . Therefore
where is the number of terms in which in this series. To prove (47), we use the Duhamel equation 
whose iteration leads to Dyson-Phillips series, see , given by
and (47) follows.
The second part of the theorem follows from Theorem 5 and Remark 7. We just need to show that the model with the transport process is conservative. We have proved  that the semigroup generated by the operator is strongly continuous and stochastic and this implies for all , which leads to where is the total mass of the system defined in (6) and therefore proving the conservativeness of the transport process.
We know, see [3, Corollary 6.4], that under the assumptions of the previous Lemma, the Cauchy problem has a solution defined on the product set whenever and given by where for all with defined in Theorem 4. In the same way, if is the evolution semigroup associated with , (see (14)) then for all with defined in Lemma 2. Because is bounded, it follows that the perturbed Cauchy problem also has a solution defined on the product set whenever . We can state the following.
Lemma 9. Let fix and consider the families and defined in Lemma 8. The family defined by
exists for all and forms a positive, -semigroup on conserving the norm .
In the same way the family defined by exists for all and forms a forward propagator on conserving the norm .
Proof. Let ; from (27) and (28) we have
Hence, the family is a Cauchy sequence in the Banach space and therefore convergent and its limits uniform in exist as . Furthermore
where we have used the continuity of on and the fact that conserves the norm . Thus conserves the norm :
is a semigroup, in fact
(a) In the same way as previously, we show, using the definition of , that Hence This convergence, being strong in finally yields :
(b) since for all .
(c) By a similar way we show that since the limits (62) exist uniformly in .
(d) is positive since from the definition of and [16, Corollary 5.11], there exists a subsequence so that Because is positive then if , this pointwise limit gives and the results follow. The proof of the second part of this lemma is very similar to the proof of the first part with the additional note that the limit (63) is uniform in and , which concludes the proof of the lemma.
The above discussions allow us to state the following existence result for an approximated discrete and nonautonomous fragmentation model in a moving medium.
Theorem 10. Let . Under the assumptions of Lemma 8, the families defined in (62) and (63) are, respectively, an evolution semigroup and a forward propagator generated by the operator , defining the Cauchy problem (61). Furthermore the solution given, for all initial data , by satisfies the perturbed Cauchy problem (61).
5. Existence Results: Discussion and Concluding Remark
Now what happens if we tend . We know, via (44), that the perturbed Cauchy problem (61) tends to discrete and nonautonomous fragmentation model in a moving medium (3) as . For this reason, the solution obtained in Theorem 10 can serve to approximate a solution for the Cauchy problem (3). The results obtained here, where we had to deal with a two parameter family of bounded linear operators, improve the preceding ones [2, 3] where the two processes involved in the system, namely, transport and nonautonomous fragmentation, were treated separately. However, the problem of characterizing the full generator is still an open problem for this type of perturbed nonautonomous and transport model.
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