Global Analysis of a Discrete Nonlocal and Nonautonomous Fragmentation Dynamics Occurring in a Moving Process

Abstract

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 𝑛 (with position 𝑥) over a short period of time and is given by the following differential equation as presented in [1]: 𝜕𝑝()𝜔()𝑝()+𝜕𝑡𝑡,𝑥,𝑛=−div𝑥,𝑛𝑡,𝑥,𝑛_{∞𝑚=𝑛+1}𝑎()𝑏()𝑝()()𝑝(),𝑝𝜏𝑡,𝑥,𝑚𝑡,𝑥,𝑛,𝑚𝑡,𝑥,𝑚−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³, (1) where 𝑝(𝑡,𝑥,𝑛) is the particle mass distribution function with respect to the mass 𝑛 at the position 𝑥∈ℝ³ and time 𝑡, (𝑝_𝜏0(𝑥,𝑛)≡𝑝_𝑛,𝜏0(𝑥) is the mass distribution at some fixed time 𝜏₀≥0), 𝑏(𝑡,𝑥,𝑛,𝑚)≡𝑏_𝑛,𝑚(𝑡,𝑥) is the distribution of particle masses 𝑛 and position 𝑥, spawned by the fragmentation of a particle of mass 𝑚 at time 𝑡, 𝑇∈ℝ, and 𝑎(𝑡,𝑥,𝑛)≡𝑎_𝑛(𝑡,𝑥) is the evolutionary time-dependent fragmentation rate, that is, the rate at which mass 𝑛 particles at position 𝑥 break up at a time 𝑡. The velocity 𝜔=𝜔(𝑥,𝑛) of the transport is supposed to be a known quantity, depending on the size 𝑛 of aggregates and their position 𝑥.

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 [2] 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 [3], 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 [4] used evolution semigroups approach which allows them to build on the substochastic semigroup theory and obtain an equivalent result as in [3]. In the analysis of the book by Kato [5] and later improved by Da Prato and Grisvard [6], it is generally assumed that the generators 𝐴(𝑡) and 𝐵(𝑡) involved in the perturbation are of class 𝒢(1,0). This condition is modified in [7] 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 [10], we focus on the case where after clusters fragmentation, new originating groups have different centers distributed according to a given probabilistic law ℎ(⋅,𝑛,𝑚,𝑦). This is the probability density that after a break up of an 𝑚-aggregate (with the center at 𝑦), the new formed 𝑛-group will be located at 𝑥. Therefore _ℝ3ℎ𝑥,𝑛,𝑚,𝑦𝑑𝑥=1, (2) and the system (1) becomes 𝜕𝑝()𝜔()𝑝()+𝜕𝑡𝑡,𝑥,𝑛=−div𝑥,𝑛𝑡,𝑥,𝑛_{∞𝑚=𝑛+1ℝ3}𝑎𝑏𝑝()𝑝(),𝑝𝜏𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³. (3) Since a group of size 𝑚≤𝑛 cannot split to form a group of size 𝑛, we require 𝑏_𝑛,𝑚(𝑡,𝑥)=0 (4) at any time 𝑡 and position 𝑥 for all 𝑚≤𝑛. We also set 𝑎₁(𝑡,𝑥)=0,^𝑛−1_𝑚=1𝑚𝑏_𝑚,𝑛(𝑡,𝑥)=𝑛(𝑛=2,3,…), (5) meaning that a cluster of size one cannot split and the sum of all individuals obtained by fragmentation at a position 𝑥 of an 𝑛-group is equal to 𝑛 all the time 𝑡. The second term on the right-hand side of (3) describes the increase in the number of particles of size 𝑛 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 𝑛 due to the fission of groups of the same size (the loss due to the fragmentation). The space variable 𝑥 is supposed to vary in the whole of ℝ³.

2. Approximation and Analysis of the Fragmentation Operator

Since 𝑝=𝑝(𝑡,𝑥,𝑛) is the density of 𝑛-groups at the position 𝑥 and time 𝑡 and that total mass 𝑈(𝑡)=^∞_𝑛=1𝑛_ℝ3𝑝(𝑡,𝑥,𝑛)𝑑𝑥 (6) is expected to be a conserved quantity, the most appropriate Banach space to work in is the space 𝒳₁∶=𝐿₁ℝ³×ℕ,𝑑𝜇𝑑𝑚₁, (7) where ℕ is equipped with the weighed counting measure 𝑑𝑚₁ with weight 𝑛 and 𝑑𝜇=𝑑𝑥 is the Lebesgue measure in ℝ³. Note that elements’ norm of 𝒳₁ represents the total mass (or total number of individuals) of the system.

Now we recast (3) as the nonautonomous abstract Cauchy problem in 𝒳₁: 𝜕𝑝()=)(𝑡)𝑝(),𝑝𝜏𝜕𝑡𝑡,𝑥,𝑛𝒟𝑝(𝑡,𝑥,𝑛+𝑄𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³, (8) or in the compact form 𝜕𝐩(𝑡)(𝑡)(𝑡)𝐩(𝑡),|𝜕𝑡=𝐃𝐩+𝒬𝐩(𝑡)_𝑡=𝜏0=𝐩_𝜏0, (9) where 𝐩(𝑡) is the vector 𝐩(𝑡)=(𝑝(𝑡,𝑥,𝑛))^∞_𝑛=1, 𝐩_𝜏0 the mass distribution vector (𝑝_𝑛(𝜏₀,𝑥))^∞_𝑛=1 at the fixed time 𝜏₀≥0 and position 𝑥, and 𝒬(𝑡) the nonautonomous fragmentation operator defined by 𝒬(𝑡)𝐩(𝑡)=𝑄(𝑡)𝑝(𝑡)(𝑡,𝑥,𝑛)^∞_𝑛=1∶=^∞_{𝑚=𝑛+1ℝ3}𝑎𝑏𝑝()𝑝(𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛)^∞_𝑛=1, (10) where 𝑄(𝑡) is seen as the pointwise operation 𝜓(𝑡,𝑥,𝑛)⟼^∞_{𝑚=𝑛+1ℝ3}𝑎𝑏𝜓()𝜓()𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑛𝑑𝑦−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛 (11) defined on the set of measurable functions. 𝑄(𝑡) is defined by 𝑄(𝑡)=𝔔(𝑡) and represents the realization of 𝔔(𝑡) on the domain 𝐷(𝑄(𝑡))=𝑝∈𝒳₁;𝔔(𝑡)𝑝(𝑡)∈𝒳₁, (12) with 𝔔(𝑡)𝑝(𝑡) given by (11). The transport matrix 𝐃 is given by 𝐃𝐩∶=(𝒟𝑝(𝑡,𝑥,𝑛))^∞_𝑛=1 with 𝒟𝑝(𝑡,𝑥,𝑛)=−div𝜔(𝑥,𝑛)𝑝(𝑡,𝑥,𝑛). (13) 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 [11] or propagator [7]). A two parameter family of bounded linear operators 𝑈(𝑡,𝜏), 0≤𝜏≤𝑡≤𝑇, on a Banach space is called forward propagator or evolution system if the following conditions are respected:(i)𝑈(𝜏,𝜏)=𝐼 for 0≤𝜏≤𝑇;(ii)𝑈(𝑡,𝑟)𝑈(𝑟,𝜏)=𝑈(𝑡,𝜏) for 0≤𝜏≤𝑟≤𝑡≤𝑇;(iii)(𝑡,𝜏)→𝑈(𝑡,𝜏) is strongly continuous for 0≤𝜏≤𝑡≤𝑇.
Recall that [7] the forward propagator 𝑈(𝑡,𝜏), 0≤𝜏≤𝑡≤𝑇 can be associated with the so-called evolution semigroup (𝐺_𝑡(𝑠))_𝑠≥0 defined in 𝒳₁; that is, if for any fixed 𝑡∈ℑ=[0,𝑇], the operator 𝑄(𝑡) generates a forward propagator 𝑈(𝑡,𝜏), 0≤𝜏≤𝑡≤𝑇, then this propagator defines a 𝐶₀ semigroup (𝐺_𝑡(𝑠))_𝑠≥0 given by the relation 𝐺_𝑡(𝑠)𝑝(𝜎)=(𝜎−𝑠)𝜒_ℑ𝑈(𝜎,𝜎−𝑠)𝑝(𝜎−𝑠), (14) where 𝜒_ℑ is the characteristic function of ℑ=[0,𝑇], 𝑝∈𝒳₁, and 𝜎∈ℑ. In the following pages, when we say 𝑄(𝑡) is the generators of 𝐶₀ semigroups in 𝒳₁, we mean that 𝑄(𝑡) generates a propagator which defines a 𝐶₀ semigroup in 𝒳₁ satisfying the relation (14).
We introduce, for any given 𝑘∈ℕ, the projection operator 𝑞_𝑘 defined for a function 𝑔∈𝒳₁ as 𝑞_𝑘𝑔)=𝑔()(𝑥,𝑛𝑥,𝑛,𝑘>𝑛>0,𝑥∈ℝ³0,otherwise. (15) The space 𝑋_𝑘=𝑔∈𝒳₁∶𝑔(𝑥,𝑛)≡0onℝ³×(𝑘,∞) (16) is therefore a closed subspace of 𝒳₁ on which the projection operator 𝑞_𝑘 acts. Let us associate with the fragmentation model 𝑑𝑝(𝑡)(𝑡)𝑝(𝑡),𝑝𝜏𝑑𝑡=𝑄₀=𝑝_𝜏0,0≤𝜏₀<𝑡≤𝑇, (17) the following truncated version: 𝑑𝑝(𝑡)(𝑡)𝑞𝑑𝑡=𝑄_𝑘𝑝(𝑡),𝑝𝜏₀=𝑝_𝜏0,0≤𝜏₀<𝑡≤𝑇, (18) where 𝑄(𝑡) is represented by (11). We set 𝑄_𝑘(𝑡)=𝑄(𝑡)𝑞_𝑘, then 𝑄_𝑘(𝑡) can be decomposed as 𝑄_𝑘(𝑡)=𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡) where the loss and the gain fragmentation operators 𝐴_𝑘(𝑡) and 𝐵_𝑘(𝑡) are given by 𝐴_𝑘(𝑡)𝑔(𝑡,𝑥,𝑛)=𝐴(𝑡)𝑞_𝑘𝑔()()𝑔(),𝐵𝑡,𝑥,𝑛=−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛_𝑘(𝑡)𝑔(𝑡,𝑥,𝑛)=𝐵(𝑡)𝑞_𝑘𝑔()=𝑡,𝑥,𝑛_{𝑘𝑚=𝑛+1ℝ3}𝑎𝑏𝑔𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦, (19) where 𝐴(𝑡) and 𝐵(𝑡) are expressed as 𝐴_𝑘(𝑡)(𝑡),𝐵(𝑡)𝑔()==𝐴𝑡,𝑥,𝑛^∞_{𝑚=𝑛+1ℝ3}𝑎𝑏𝑔𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦. (20)(21) Thus, for all 𝑔∈𝒳₁, 𝑘∈ℕ and (𝑡,𝑥,𝑛)∈ℑ×ℝ³×ℕ,𝑄_𝑘(𝑡)𝑔()=𝑡,𝑥,𝑛_{𝑘𝑚=𝑛+1ℝ3}𝑎𝑡,𝑦,𝑚𝑏𝑡,𝑦,𝑛,𝑚ℎ𝑥,𝑛,𝑚,𝑦𝑔𝑡,𝑦,𝑚−𝑎(𝑡,𝑥,𝑛)𝑔(𝑡,𝑥,𝑛),𝑘>𝑛>00,otherwise, (22)with 𝐷𝑄_𝑘(𝑡)=𝑔∈𝒳₁;𝑄_𝑘(𝑡)𝑔∈𝑋_𝑘. (23) We assume that for 𝑡 fixed in ℝ, there are two constants 0<𝜃₁ and 𝜃₂ such that 𝜃₁𝛼(𝑛,𝑡)≤𝑎_𝑡(𝑥,𝑛)=𝑎(𝑡,𝑥,𝑛)≤𝜃₂𝛼()()𝑛,𝑡foralmostall𝑥,𝑛∈ℝ³×ℕ, (24) where 𝛼(𝑛,𝑡)∈𝐿_∞,loc(ℝ₊²) is a real-valued function which can depend on 𝑛 and 𝑡 but is independent of the state variable 𝑥. This obviously implies that for any 𝑘∈ℕ there exists a positive Θ_𝑡,𝑘 such that esssup_{ℝ3×(0,𝑘)}𝑎_𝑡(𝑥,𝑛)≤Θ_𝑡,𝑘. (25) Moreover the sequence {Θ_𝑡,𝑘}_𝑘∈ℕ (𝑡 fixed in ℝ) is bounded in the following way: esssup_{0≤𝑡≤𝑇}Θ_𝑡,𝑘≤Θ_𝑇,𝑘<∞. (26)

Lemma 2. For 𝑡 fixed in ℑ, and 𝑘∈ℕ, there is a positive uniformly continuous semigroup of contractions on 𝑋_𝑘⊂𝒳₁, say (𝐺_𝑡,𝑘(𝑠))_𝑠≥0 generated by operator 𝑄_𝑘(𝑡) such that (𝐺_𝑡,𝑘(𝑠))_𝑠≥0 is conservative on (𝑋_𝑘)₊ and given by 𝐺_𝑡,𝑘(𝑠)=𝐼+^∞_𝑗=1𝑠^𝑗(𝑄(𝑡))^𝑗𝑞𝑗!^𝑘. (27) Moreover, for any 𝑟≥𝑘, 𝑠≥0, 𝑞_𝑘𝐺_𝑡,𝑟(𝑠)𝑞_𝑘=𝐺_𝑡,𝑟(𝑠). (28)

Proof. Let us fix 𝑡 in ℑ. The operator 𝐴_𝑘(𝑡) is bounded by (25). Changing the order of summation by the Tonelli’s theorem, for every 𝑔∈𝑋_𝑘, 𝐵_𝑘(𝑡)𝑔₁=_ℝ3^∞_𝑛=1𝑛^𝑘_{𝑚=𝑛+1ℝ3}𝑎𝑏𝑔=𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦𝑑𝑥_{ℝ3∞𝑛=1}𝑛^𝑘_𝑚=𝑛+1𝑎𝑏𝑔=𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚𝑡,𝑦,𝑚𝑑𝑦_{ℝ3𝑘𝑚=2}𝑎𝑡,𝑦,𝑚𝑔𝑡,𝑦,𝑚^∞_𝑛=1=𝑛𝑏𝑡,𝑦,𝑛,𝑚𝑑𝑦_{ℝ3𝑘𝑚=2}𝑎𝑡,𝑦,𝑚𝑔𝑡,𝑦,𝑚^𝑚−1_𝑛=1=𝑛𝑏𝑡,𝑦,𝑛,𝑚𝑑𝑦_{ℝ3𝑘𝑚=2}𝑔=𝑚𝑎𝑡,𝑦,𝑚𝑡,𝑦,𝑚𝑑𝑦_{ℝ3𝑘𝑚=1}𝑔≤𝑚𝑎𝑡,𝑦,𝑚𝑡,𝑦,𝑚𝑑𝑦_{ℝ3∞𝑚=1}𝑔=𝐴𝑚𝑎𝑡,𝑦,𝑚𝑡,𝑦,𝑚𝑑𝑦_𝑘(𝑡)𝑔₁=Θ_𝑡,𝑘𝑔₁<∞, (29) where we have used (2) and (5), respectively. Then 𝐵_𝑘(𝑡) is also bounded. Hence 𝑄_𝑘(𝑡) generates a uniformly continuous semigroup. We denote this semigroup by (𝐺_𝑡,𝑘(𝑠))_𝑠≥0. Clearly, 𝐴_𝑘(𝑡) generates a positive semigroup of contractions and 𝐵_𝑘(𝑡) is a positive operator. Moreover, the above calculations also imply that 𝐷(𝐵_𝑘(𝑡))⊃𝐷(𝐴_𝑘(𝑡)) and ^∞_𝑛=1ℝ3𝑛𝐴_𝑘(𝑡)𝑔(𝑡,𝑥,𝑛)+𝐵_𝑘(𝑡)𝑔(=𝑡,𝑥,𝑛)𝑑𝑥_{𝑘𝑛=1ℝ3}𝑛𝐴_𝑘(𝑡)𝑔(𝑡,𝑥,𝑛)+𝐵_𝑘(𝑡)𝑔(𝑡,𝑥,𝑛)𝑑𝑥≤0 (30) for all 𝑔∈(𝐷(𝐴_𝑘(𝑡)))₊ with 𝐷𝐴_𝑘(𝑡)=𝑔∈𝒳₁;−𝑎_𝑡𝑔∈𝑋_𝑘,𝐷𝐵_𝑘(𝑡)=𝑔∈𝒳₁;𝐵_𝑘(𝑡)𝑔∈𝑋_𝑘. (31) Thus the assumptions of Kato’s theorem in 𝐿₁-space [12, Corollary 5.17] hold. We essentially note that for each fixed 𝑡 the operator 𝑄_𝑘(𝑡) becomes independent of time [3, Lemma 2.1] and Kato’s theorem is immediately applicable. Therefore there is an extension 𝑄̆_𝑘(𝑡) of 𝑄_𝑘(𝑡) which generates a substochastic semigroup. Because 𝑎_𝑡(𝑥,𝑛) is bounded in ℝ³×(0,𝑘), this substochastic semigroup is conservative and it follows that 𝑄̆_𝑘(𝑡)=𝑄_𝑘(𝑡), where 𝑄_𝑘(𝑡) is the closure of 𝑄_𝑘(𝑡). Since 𝑄_𝑘(𝑡) generates a uniformly (and hence strongly) continuous semigroup, 𝑄_𝑘(𝑡) is a closed operator. Therefore we have that 𝑄̆_𝑘(𝑡)=𝑄_𝑘(𝑡); consequently, the uniformly continuous semigroup (𝐺_𝑡,𝑘(𝑠))_𝑠≥0 is a positive strongly continuous semigroup of contractions and furthermore (𝐺_𝑡,𝑘(𝑠))_𝑠≥0 is honest.
The proof of (27) is clear since we can use the usual power series definition to define 𝐺_𝑡,𝑘(𝑠)=exp(𝑠𝑄_𝑘(𝑡)). By induction, (𝑄_𝑘(𝑡))^𝐽=(𝑄(𝑡)𝑞_𝑘)^𝐽=(𝑄(𝑡))^𝐽𝑞_𝑘 for 𝐽=1,2,…, from which the exponential formula yields (27).
To prove (28), we have that 𝐵(𝑡)𝑞_𝑘𝑔=𝑞_𝑘𝐵(𝑡)𝑞_𝑘𝑔 on 𝐵(𝑡)(0_ℝ³,𝑘)×[0,𝑘] since for 𝑘≥𝑛≥0, 𝐵(𝑡)𝑞_𝑘𝑔(𝑡,𝑥,𝑛) is given by (21) and 𝐵(𝑡)𝑞_𝑘𝑔(𝑡,𝑥,𝑛)=0 for 𝑘<𝑛. Moreover, it is obvious that 𝐴(𝑡)𝑞_𝑘𝑔=𝑞_𝑘𝐴(𝑡)𝑞_𝑘𝑔; hence we have also (𝐴(𝑡)+𝐵(𝑡))𝑞_𝑘=𝐴(𝑡)𝑞_𝑘+𝐵(𝑡)𝑞_𝑘=𝑞_𝑘𝐴(𝑡)𝑞_𝑘+𝑞_𝑘𝐵(𝑡)𝑞_𝑘=𝐴_𝑘(𝑡)+𝐵_𝑘(𝑡)=𝑄_𝑘(𝑡). (32) Next, by 𝑞_𝑘𝑞_𝑟=𝑞_𝑟𝑞_𝑘=𝑞_𝑘 we have 𝑞_𝑘𝑄_𝑟(𝑡)𝑞_𝑘=𝑞_𝑘𝑞_𝑟𝑄(𝑡)𝑞_𝑟𝑞_𝑘=𝑞_𝑘𝑄(𝑡)𝑞_𝑘=𝑄_𝑘(𝑡); (33) if we assume, by induction, that 𝑞_𝑘(𝑄_𝑟(𝑡))^𝑗−1𝑞_𝑘=(𝑄_𝑘(𝑡))^𝑗−1, then 𝑞_𝑘𝑄_𝑟(𝑡)^𝑗𝑞_𝑘=𝑞_𝑘𝑄_𝑟(𝑡)^𝑗−1𝑄_𝑟(𝑡)𝑞_𝑘=𝑞_𝑘𝑄_𝑟(𝑡)^𝑗−1𝑞_𝑟𝑄(𝑡)𝑞_𝑟𝑞_𝑘=𝑞_𝑘𝑄_𝑟(𝑡)^𝑗−1𝑞_𝑟𝑞_𝑘𝑄(𝑡)𝑞_𝑘=𝑞_𝑘𝑄_𝑟(𝑡)^𝑗−1𝑞_𝑘𝑄_𝑘(𝑡)=𝑄_𝑘(𝑡)^𝑗. (34) Now using (27) and the the fact that 𝑄_𝑟(𝑡) is a bounded operator, the semigroup generated by 𝑄_𝑟(𝑡) is expressed by 𝑞_𝑘𝐺_𝑡,𝑟(𝑠)𝑞_𝑘=^∞_𝑛=0𝑠^𝑗𝑞_𝑘𝑄_𝑟(𝑡)^𝑗𝑞_𝑘=𝑛!_∞𝑛=0𝑠^𝑗𝑄_𝑘(𝑡)^𝑗𝑗!=𝐺_𝑡,𝑟(𝑠), (35) which concludes the lemma.

Next we assume that 𝑎(𝑡,𝑥,𝑛) satisfies the Lipschitz condition |𝑎(𝑡,𝑥,𝑛)−𝑎(𝜎,𝑥,𝑛)|≤|𝑡−𝜎|Λ(𝑥,𝑛),𝑡,𝜎∈ℑ, (36) where Λ(𝑥,𝑛)≥0 together with Λ(𝑥,𝑛)≤Θ_𝑘; for all 𝑘≥𝑛>0 we state the following lemma.

Lemma 3. The function 𝑡→𝑄_𝑘(𝑡) is continuous in the uniform operator topology for each 𝑘 fixed in ℕ.

Proof. Using Fubini’s theorem and assumption (36) yields 𝑄_𝑘(𝑡)𝑔−𝑄_𝑘(𝜎)𝑔₁≤𝑘Θ_𝑘|𝑡−𝜎|𝑔₁,∀𝑔∈𝒳₁, (37) and the result follows.

Making use of (14) and Lemma 2, there is a forward propagator, say {𝑈_𝑘(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} defined in 𝑋_𝑘⊂𝒳₁ which is associated with the evolution semigroup (𝐺_𝑡,𝑘(𝑠))_𝑠≥0, 𝑡∈ℑ. The propagator {𝑈_𝑘(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} shares certain properties with the family of semigroups (𝐺_𝑡,𝑘(𝑠))_𝑠≥0, 𝑡∈ℑ, as stated in the following theorem and proved in [3, Theorem 4.1].

Theorem 4 (see [11]). For each 𝑘∈ℕ, the forward propagator {𝑈_𝑘(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} generated by the family of infinitesimal generators {𝑄_𝑘(𝑡)}_𝑡∈ℑ have the following properties:(1)𝑈_𝑘(𝑡,𝜏) is positive;(2)‖𝑈_𝑘(𝑡,𝜏)𝑔‖₁=‖𝑔‖₁, for all 𝑔∈𝒳₁;(3)∑^∞_𝑛=0𝑛[𝑈_𝑘(𝑡,𝜏)𝑔](𝑡,𝑥,𝑛)=∑^∞_𝑛=0𝑛𝑔(𝑡,𝑥,𝑛), for all 𝑔∈𝒳₁, 𝑥∈ℝ³;(4)(𝜕/𝜕𝑡)𝑈_𝑘(𝑡,𝜏)=𝑄_𝑘(𝑡)𝑈_𝑘(𝑡,𝜏), 0≤𝜏≤𝑡≤𝑇;(5)(𝜕/𝜕𝜏)𝑈_𝑘(𝑡,𝜏)=−𝑈_𝑘𝑄_𝑘(𝑡,𝜏), 0≤𝜏≤𝑡≤𝑇.

Theorem 5. The truncated problem (18) has a unique, strongly continuously differentiable, positive, mass-conserving solution for all initial data 𝑝(𝜏₀)=𝑝_𝜏0∈𝑋_𝑘. The solution is given by 𝑝(𝑡)=𝑈_𝑘(𝑡,𝜏₀)𝑝_𝜏0  (0≤𝜏₀≤𝑡≤𝑇).

Proof. This theorem is an immediate consequence of Lemmas 2 and 3 and Theorem 4 associated with [11, Theorem 5.1].

Remark 6. According to [11, Definition  2.1, page 130, Remark 3.2, page 138], for each fixed 𝑘>0, the family of infinitesimal generators {𝑄_𝑘(𝑡)}_𝑡∈ℑ of 𝐶₀-semigroups on 𝑋_𝑘 is stable with stability constants 1 and 0. Moreover we have from Theorem 4𝑈_𝑘(𝑡,𝜏)₁≤1,for0≤𝜏≤𝑡≤𝑇. (38)

3. Cauchy Problem for the Transport Model in ℝ³×ℕ

We consider the Cauchy problem for the transport equation 𝜕𝑝()𝜔()𝑝(),𝑝()=𝜕𝑡𝑡,𝑥,𝑛=−div𝑥,𝑛𝑡,𝑥,𝑛0,𝑥,𝑛𝑝^𝑜_𝑛(𝑥),𝑛=1,2,3,…, (39) in the space 𝒳₁.

We consider the function 𝜔∶ℝ³×ℕ→ℝ³ defined by (𝑥,𝑛)→𝜔(𝑥,𝑛) and recall that 𝒟 in (13) is the expression appearing on the right-hand side of (39). Then )𝜔()𝑝()=((𝑝()()⋅().𝐷𝑝(𝑡,𝑥,𝑛∶=−div𝑥,𝑛𝑡,𝑥,𝑛∇⋅𝜔𝑥,𝑛))𝑡,𝑥,𝑛+𝜔𝑥,𝑛∇𝑝𝑡,𝑥,𝑛 (40) We assume that (H1)𝜔 is divergence-free;(H2)𝜔 globally Lipschitz continuous; then div𝜔(𝑥,𝑛)∶=∇⋅𝜔(𝑥,𝑛)=0 and (40) becomes 𝒟𝑝(𝑡,𝑥,𝑛)∶=𝜔(𝑥,𝑛)⋅∇𝑝(𝑡,𝑥,𝑛). (41) We define in 𝒳₁ the operators (𝒟,𝐷(𝒟)) as )=),)()𝐷(𝒟)𝒟𝑝(𝑡,𝑥,𝑛𝒟𝑝(𝑡,𝑥,𝑛with𝒟𝑝(𝑡,𝑥,𝑛representedby41∶=𝑝∈𝒳₁,𝜔⋅∇𝑝𝒳₁,𝑛∈ℕ. (42)

Remark 7. With the assumptions (H1)-(H2), it is proven [2] that 𝒟 is the generator of a strongly continuous stochastic semigroup, say (𝐺_𝒟(𝑡))_𝑡≥0 and ‖𝐺_𝒟(𝑡)‖≤1=1𝑒^0𝑡. Then 𝒟∈𝒢(1,0) and due to Hille Yosida’s characterization, (𝒟,𝐷(𝒟)) is a closed and densely defined operator satisfying for the resolvent set 𝜌(𝒟)⊃[0,∞) and ‖𝜆𝑅(𝜆,𝒟)‖₁≤𝐾 for some constant 𝐾>0 and all 𝜆>0. Furthermore, (i)for any 𝑔∈𝒳₁, lim_𝜆→∞𝜆𝑅(𝜆,𝒟)𝑔=𝑔; (43)(ii)𝒟𝑅(𝜆,𝒟) are bounded operators and for any 𝑔∈𝐷(𝒟), lim_𝜆→∞𝜆𝒟𝑅(𝜆,𝒟)𝑔=𝒟𝑔. (44)

4. Perturbed Approximated Problem

Let 𝒟_𝜆=𝜆𝒟𝑅(𝜆,𝒟); Yosida [13] was the first to use the bounded operators 𝒟_𝜆 to approximate the unbounded operator 𝒟, for which we can define semigroups via the exponential formula 𝑒^𝑡𝒟=𝐼+𝑡𝒟+𝑡1!²𝒟²2!+⋯. (45) We exploit this idea to analyze the following approximated problem associated with (3): 𝜕𝑝()𝜕𝑡𝑡,𝑥,𝑛=𝑄_𝑘(𝑡)𝑝(𝑡,𝑥,𝑛)+𝒟_𝜆𝑝),𝑝𝜏(𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³. (46)

Lemma 8. Let each 𝑘 be fixed in ℕ and any 𝜆>0. Under the assumptions (5), (25), (26), (36), (H1) and (H2), the operator (𝑇_𝑘,𝜆(𝑡)=𝑄_𝑘(𝑡)+𝒟_𝜆,𝐷(𝑄_𝑘(𝑡))), (0≤𝑡≤𝑇) appearing in approximated Cauchy problem (46) is a stable generator, with the stability constants 1 and ‖𝒟_𝜆‖₁, of a forward propagator {𝕌_𝑘,𝜆(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} associated with an evolution semigroup (𝔾_{𝑡,𝑘,𝜆}(𝑠))_𝑠≥0, positive, conserving the norm ‖⋅‖₁ and given by 𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔=𝐺_𝑡,𝑘(𝑠)𝑔+^∞_𝑖=1𝐺^𝑖_𝜆(𝑠)𝑔, (47) where 𝐺^𝑖_𝜆(𝑠)𝑔=^𝑠₀𝐺_𝑡,𝑘(𝑠−𝑣)𝑔𝒟_𝜆𝐺_𝜆^𝑖−1(𝑣)𝑔𝑑𝑣,𝑔∈𝑋_𝑘 (48) with (𝐺_𝑡,𝑘(𝑠))_𝑠≥0 defined in Lemma 2.
Furthermore the problem (46) has a unique, strongly continuously differentiable, positive, mass-conserving solution for all initial data 𝑝(𝜏₀)=𝑝_𝜏0∈𝑋_𝑘. The solution is given by 𝑝(𝑡)=𝕌_𝑘,𝜆(𝑡,𝜏₀)𝑝_𝜏0  (0≤𝜏₀≤𝑡≤𝑇).

Proof. Let us fix 𝑘∈ℕ and 𝜆>0. The fact that 𝑇_𝑘,𝜆(𝑡) is the generator of a forward propagator comes from Lemma 2, the Bounded perturbation theorem, and the remark (14). By Remark 6 we have 𝜌(𝑄_𝑘(𝑡))⊃(0,∞). If 𝜈>‖𝒟_𝜆‖₁, it is obvious that 𝜈∈𝜌(𝑇_𝑘,𝜆(𝑡)) (bounded perturbation) and the resolvent satisfies 𝑅𝜈,𝑇_𝑘,𝜆(𝑡)=^∞_𝑖=1𝑅𝜈,𝑄_𝑘(𝑡)𝒟_𝜆𝑅𝜈,𝑄_𝑘(𝑡)^𝑖. (49) Henceforth, for any finite sequence 0≤𝑡₁≤𝑡₂≤⋯≤𝑡_𝑟≤𝑇, 𝑟=1,2,…, we have ^𝑟_𝑛=1𝑅𝜈,𝑇_𝑘,𝜆𝑡_𝑛=^𝑟_𝑛=1^∞_𝑖=1𝑅𝜈,𝑄_𝑘𝑡_𝑛𝒟_𝜆𝑅𝜈,𝑄_𝑘𝑡_𝑛⁻¹. (50) The development of the right-hand side of (50) yields a series with the general term in the form 𝑅𝜈,𝑄_𝑘𝑡_𝑚𝒟_𝜆𝑅𝜈,𝑄_𝑘𝑡_𝑚^𝑖𝑚⋯𝑅𝑣,𝑄_𝑘𝑡₁𝒟_𝜆𝑅𝜈,𝑄_𝑘𝑡₁^𝑖1, (51) where 𝑖_𝑛≥0. If ∑^𝑟_𝑛=1𝑖_𝑛=𝑖; then using the stability of the family 𝑄_𝑘(𝑡)_{0≤𝜏0≤𝑡≤𝑇}, we estimate (51) by ‖𝒟_𝜆‖₁𝜈^{−𝑖−𝑟}. Therefore ^𝑟_𝑛=1𝑅𝜈,𝑇_𝑘,𝜆𝑡_𝑛1≤𝜈^∞−𝑟_𝑖=1^𝑖_𝑟𝒟_𝜆1𝜈^−1−𝑖=𝜈−𝒟_𝜆1⁻¹, (52) where (^𝑖_𝑟) is the number of terms in which ∑^𝑟_𝑛=1𝑖_𝑛=𝑖 in this series. To prove (47), we use the Duhamel equation [14] 𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔=𝐺_𝑡,𝑘(𝑠)𝑔+^𝑠₀𝐺_𝑡,𝑘(𝑠−𝑣)𝑔𝒟_𝜆𝔾_{𝑡,𝑘,𝜆}(𝑣)𝑔𝑑𝑣,𝑔∈𝑋_𝑘 (53) whose iteration leads to Dyson-Phillips series, see [15], given by 𝔾_{𝑡,𝑘,𝜆}(𝑠)=^∞_𝑖=0𝐺^𝑖_𝜆(𝑠),with𝐺⁰_𝜆(𝑠)𝑔=𝐺_𝑡,𝑘(𝑠)𝑔 (54) 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 [2] that the semigroup generated by the operator 𝒟 is strongly continuous and stochastic and this implies 0=_ℝ3×ℕ𝒟𝑝𝑑𝜇𝑑𝑚₁,=_ℝ3^∞_𝑛=1𝑛𝒟𝑝(𝑡,𝑥,𝑛)𝑑𝑥, (55) for all 𝑝∈𝐷(𝒟)𝑡≥0, which leads to 𝑑𝑈(𝑡)=𝑑𝑑𝑡𝑑𝑡^∞_𝑛=1𝑛_ℝ3𝑝()=𝑡,𝑥,𝑛𝑑𝑥_∞𝑛=1𝑛_ℝ3𝜕_𝑡𝑝()=𝑡,𝑥,𝑛𝑑𝑥_∞𝑛=1𝑛_ℝ3𝒟𝑝(𝑡,𝑥,𝑛)𝑑𝑥=0, (56) where 𝑈(𝑡) 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 𝜕𝑝()=𝜕𝑡𝑡,𝑥,𝑛^∞_{𝑚=𝑛+1ℝ3}𝑎𝑏𝑝()𝑝(),𝑝𝜏𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³ (57) has a solution 𝑝 defined on the product set [0,∞)×ℝ×ℕ whenever 𝑝_𝜏0∈𝒳₁ and given by 𝑝(𝑡,𝑥,𝑛)=𝑈𝑡,𝜏₀𝑝_𝜏0(𝑥,𝑛),𝑛∈ℕ,0≤𝜏₀<𝑡, (58) where 𝑈(𝑡,𝜏)𝑔=lim_𝑘→∞𝑈_𝑘(𝑡,𝜏)𝑔 (59) for all 𝑔∈𝒳₁ with {𝑈_𝑘(𝑡,𝜏)}_{0≤𝜏<𝑡≤𝑇} defined in Theorem 4. In the same way, if (𝐺_𝑡(𝑠))_𝑠≥0 is the evolution semigroup associated with 𝑈(𝑡,𝜏), (see (14)) then 𝐺_𝑡(𝑠)𝑔=lim_𝑘→∞𝐺_𝑡,𝑘(𝑠)𝑔 (60) for all 𝑔∈𝒳₁ with {𝐺_𝑡,𝑘(𝑠)}_𝑠≥0 defined in Lemma 2. Because 𝒟_𝜆 is bounded, it follows that the perturbed Cauchy problem 𝜕𝑝()𝜕𝑡𝑡,𝑥,𝑛=𝒟_𝜆𝑝()+𝑡,𝑥,𝑛_{∞𝑚=𝑛+1ℝ3}𝑎𝑏𝑝()𝑝(),𝑝𝜏𝑡,𝑦,𝑚𝑡,𝑦,𝑛,𝑚×ℎ𝑥,𝑛,𝑚,𝑦𝑡,𝑦,𝑚𝑑𝑦−𝑎𝑡,𝑥,𝑛𝑡,𝑥,𝑛₀,𝑥,𝑛=𝑝_𝜏0(𝑥,𝑛),0≤𝜏₀<𝑡≤𝑇,𝑛=1,2,3,…,𝑥∈ℝ³ (61) also has a solution 𝑝 defined on the product set [0,∞)×ℝ×ℕ whenever 𝑝_𝜏0∈𝒳₁. We can state the following.

Lemma 9. Let fix 𝜆>0 and consider the families (𝔾_{𝑡,𝑘,𝜆}(𝑠))_𝑠≥0 and {𝕌_𝑘,𝜆(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} defined in Lemma 8. The family (𝔾_𝑡,𝜆(𝑠))_𝑠≥0 defined by 𝔾_𝑡,𝜆(𝑠)𝑔=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔,𝑔∈𝒳₁ (62) exists for all 0≤𝑡≤𝑇 and (𝔾_𝑡,𝜆(𝑠))_𝑠≥0 forms a positive, 𝐶₀-semigroup on 𝒳₁ conserving the norm ‖⋅‖₁.
In the same way the family {𝕌_𝜆(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} defined by 𝕌_𝜆(𝑡,𝜏)𝑔=lim_𝑘→∞𝕌_𝑘,𝜆(𝑡,𝜏)𝑔,𝑔∈𝒳₁ (63) exists for all 0≤𝜏<𝑡≤𝑇 and {𝕌_𝜆(𝑡,𝜏)}_{0≤𝜏≤𝑡≤𝑇} forms a forward propagator on 𝒳₁ conserving the norm ‖⋅‖₁.

Proof. Let 𝑔∈𝒳₁; from (27) and (28) we have 𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑞_𝑟𝑔=𝔾_{𝑡,𝑟,𝜆}(𝑠)𝑔+𝑞_𝑟𝑔−𝑔,where𝑘≥𝑟. (64) Then 𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔−𝔾_{𝑡,𝑟,𝜆}(𝑠)𝑔₁≤𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔−𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑞_𝑟𝑔₁+𝑔−𝑞_𝑟𝑔₁=2𝑔−𝑞_𝑟𝑔₁⟶0,as𝑘,𝑟⟶∞. (65) Hence, the family (𝔾_{𝑡,𝑘,𝜆}(𝑠))_𝑘∈ℕ is a Cauchy sequence in the Banach space 𝒳₁ and therefore convergent and its limits uniform in 𝑠 exist as 𝔾_𝑡,𝜆(𝑠)∶𝒳₁→𝒳₁. Furthermore 𝔾_𝑡,𝜆(𝑠)𝑔₁=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔₁=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑠)𝑔₁=lim_𝑘→∞𝑔₁=𝑔₁, (66) where we have used the continuity of ‖⋅‖₁ on 𝒳₁ and the fact that {𝔾_{𝑡,𝑘,𝜆}(𝑠)} conserves the norm ‖⋅‖₁. Thus {𝔾_𝑡,𝜆(𝑠)}_𝑠≥0 conserves the norm ‖⋅‖₁:
(𝔾_{𝑡,𝑘,𝜆}(𝑠))_𝑠≥0 is a semigroup, in fact
(a) 𝔾_𝑡,𝜆(𝑠+𝑣)𝑔=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑠+𝑣)𝑔=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑠)lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(𝑣)𝑔. (67) In the same way as previously, we show, using the definition of (𝔾_𝑡,𝜆(𝑠))_𝑠≥0, that 𝔾_{𝑡,𝑘,𝜆}(𝑠)𝔾_{𝑡,𝑘,𝜆}(𝑣)𝑔−𝔾_𝑡,𝜆(𝑠)𝔾_𝑡,𝜆(𝑣)𝑔₁⟶0,as𝑘,𝑟⟶∞. (68) Hence 𝔾_𝑡,𝜆(𝑠)𝔾_𝑡,𝜆(𝑣)𝑔⟵𝔾_{𝑡,𝑘,𝜆}(𝑠)𝔾_{𝑡,𝑘,𝜆}(𝑣)𝑔=𝔾_{𝑡,𝑘,𝜆}(𝑠+𝑣)𝑔⟶𝔾_𝑡,𝜆(𝑠+𝑣)𝑔as𝑘⟶∞. (69) This convergence, being strong in 𝒳₁ finally yields 𝔾_𝑡,𝜆(𝑠)𝔾_𝑡,𝜆(𝑣)=𝔾_𝑡,𝜆(𝑠+𝑣):
(b) 𝔾_𝑡,𝜆(0)=𝐼 since 𝔾_𝑡,𝜆(0)𝑔=lim_𝑘→∞𝔾_{𝑡,𝑘,𝜆}(0)𝑔=lim_𝑘→∞𝑔=𝑔 (70) for all 𝑔∈𝒳₁.
(c) By a similar way we show that lim_𝑠→0⁺𝔾_{𝑡,𝑘,𝜆}𝑔=𝑔 since the limits (62) exist uniformly in 𝑠.
(d) {𝔾_𝑡,𝜆(𝑠)} is positive since from the definition of 𝔾_𝑡,𝜆(𝑠) and [16, Corollary 5.11], there exists a subsequence {𝔾_{𝑡,𝑘𝛿,𝜆}(𝑠)𝑔}  (𝛿=1,2,…) so that 𝔾_{𝑡,𝑘𝛿,𝜆}(𝑠)𝑔(𝑥,𝑛)⟶𝔾_𝑡,𝜆(𝑠)𝑔),()(𝑥,𝑛as𝛿⟶∞fora.e.𝑥.𝑛∈ℝ³×ℕ. (71) Because {𝔾_{𝑡,𝑘𝛿,𝜆}(𝑠)𝑔} is positive then if 𝑔>0, this pointwise limit gives 𝔾_𝑡,𝜆(𝑠)𝑔(𝑥,𝑛)≥0fora.e.(𝑥.𝑛)∈ℝ³×ℕ, (72) 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 𝑡 and 𝜏, 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 𝜆>0. 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 (𝑇_𝜆(𝑡)=𝑄(𝑡)+𝒟_𝜆,𝐷(𝑄(𝑡))), (0≤𝑡≤𝑇) defining the Cauchy problem (61). Furthermore the solution given, for all initial data 𝑝(𝜏₀)=𝑝_𝜏0∈𝒳₁, by 𝑝(𝑡,𝑥,𝑛)=𝕌_𝜆𝑡,𝜏₀𝑝_𝜏0(𝑥,𝑛),𝑛∈ℕ,0≤𝜏₀<𝑡, (73) satisfies the perturbed Cauchy problem (61).

5. Existence Results: Discussion and Concluding Remark

Now what happens if we tend 𝜆→∞!. We know, via (44), that the perturbed Cauchy problem (61) tends to discrete and nonautonomous fragmentation model in a moving medium (3) as 𝜆→∞. 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.