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

Authors: E. F. Doungmo Goufo; S. C. Oukouomi Noutchie
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 sizen (with position x) over a short period of time and is given by the following differential equation as presented in [1]:
(1)∂∂tp(t,x,n)=-div(ω(x,n)p(t,x,n))∂∂tp(t,x,n)+∑m=n+1∞‍a(t,x,m)b(t,x,n,m)p(t,x,m)∂∂tp(t,x,n)-a(t,x,n)p(t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3,
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, (pτ0(x,n)≡pn,τ0(x) is the mass distribution at some fixed time τ0≥0), b(t,x,n,m)≡bn,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)≡an(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
(2)∫ℝ3‍h(x,n,m,y)dx=1,
and the system (1) becomes
(3)∂∂tp(t,x,n)=-div(ω(x,n)p(t,x,n))+∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)p(t,y,m)dy-a(t,x,n)p(t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3.
Since a group of size m≤n cannot split to form a group of size n, we require
(4)bn,m(t,x)=0
at any time t and position x for all m≤n. We also set
(5)a1(t,x)=0,∑m=1n-1‍mbm,n(t,x)=n(n=2,3,…),
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

Sincep=p(t,x,n) is the density of n-groups at the position x and time t and that total mass
(6)U(t)=∑n=1∞‍n∫ℝ3‍p(t,x,n)dx
is expected to be a conserved quantity, the most appropriate Banach space to work in is the space
(7)𝒳1∶=L1(ℝ3×ℕ,dμdm1),
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:
(8)∂∂tp(t,x,n)=[𝒟~p](t,x,n)+Q(t)p(t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3,
or in the compact form
(9)∂∂tp(t)=Dp(t)+𝒬(t)p(t),p(t)|t=τ0=pτ0,
where p(t) is the vector p(t)=(p(t,x,n))n=1∞, pτ0 the mass distribution vector (pn(τ0,x))n=1∞ at the fixed time τ0≥0 and position x, and 𝒬(t) the nonautonomous fragmentation operator defined by
(10)𝒬(t)p(t)=([Q(t)p(t)](t,x,n))n=1∞≔(∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)p(t,y,m)dy-a(t,x,n)p(t,x,n)∑m=n+1∞‍)n=1∞,
where Q(t) is seen as the pointwise operation
(11)ψ(t,x,n)⟼∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)hhhhhhhh×h(x,n,m,y)ψ(t,y,n)dy-a(t,x,n)ψ(t,x,n)
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
(12)D(Q(t))={p∈𝒳1;𝔔(t)p(t)∈𝒳1},
with 𝔔(t)p(t) given by (11). The transport matrix D is given by Dp:=(𝒟~p(t,x,n))n=1∞ with
(13)[𝒟~p](t,x,n)=-div(ω(x,n)p(t,x,n)).
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 operatorsU(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 (Gt(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 (Gt(s))s≥0 given by the relation
(14)[Gt(s)p](σ)=(σ-s)χℑU(σ,σ-s)p(σ-s),
where χℑ 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 givenk∈ℕ, the projection operator qk defined for a function g∈𝒳1 as
(15)[qkg](x,n)={g(x,n),k>n>0,x∈ℝ30,otherwise.
The space
(16)Xk={g∈𝒳1:g(x,n)≡0onℝ3×(k,∞)}
is therefore a closed subspace of 𝒳1 on which the projection operator qk acts. Let us associate with the fragmentation model
(17)dpdt(t)=Q(t)p(t),p(τ0)=pτ0,0≤τ0<t≤T,
the following truncated version:
(18)dpdt(t)=Q(t)qkp(t),p(τ0)=pτ0,0≤τ0<t≤T,
where Q(t) is represented by (11). We set Qk(t)=Q(t)qk, then Qk(t) can be decomposed as Qk(t)=Ak(t)+Bk(t) where the loss and the gain fragmentation operators Ak(t) and Bk(t) are given by
(19)Ak(t)g(t,x,n)=A(t)qkg(t,x,n)=-a(t,x,n)g(t,x,n),Bk(t)g(t,x,n)=B(t)qkg(t,x,n)=∑m=n+1k‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)g(t,y,m)dy,
where A(t) and B(t) are expressed as
(20)Ak(t)=A(t),(21)B(t)g(t,x,n)=∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)g(t,y,m)dy.
Thus, for all g∈𝒳1, k∈ℕ and (t,x,n)∈ℑ×ℝ3×ℕ,(22)Qk(t)g(t,x,n)={∑m=n+1k‍∫ℝ3‍a(t,y,m)b(t,y,n,m)h(x,n,m,y)g(t,y,m)-a(t,x,n)g(t,x,n),k>n>00,otherwise,with
(23)D(Qk(t))={g∈𝒳1;Qk(t)g∈Xk}.
We assume that for t fixed in ℝ, there are two constants 0<θ1 and θ2 such that
(24)θ1α(n,t)≤at(x,n)=a(t,x,n)≤θ2α(n,t)iiiiiiiiiiiiiiiiiforalmostall(x,n)∈ℝ3×ℕ,
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 Θt,k such that
(25)esssupℝ3×(0,k)at(x,n)≤Θt,k.
Moreover the sequence {Θt,k}k∈ℕ (t fixed in ℝ) is bounded in the following way:
(26)esssup0≤t≤TΘt,k≤Θ-T,k<∞.Lemma 2.
Fort fixed in ℑ, and k∈ℕ, there is a positive uniformly continuous semigroup of contractions on Xk⊂𝒳1, say (Gt,k(s))s≥0 generated by operator Qk(t) such that (Gt,k(s))s≥0 is conservative on (Xk)+ and given by
(27)Gt,k(s)=I+∑j=1∞‍[(sj(Q(t))jj!)]qk.
Moreover, for any r≥k, s≥0,
(28)qkGt,r(s)qk=Gt,r(s).Proof.
Let us fixt in ℑ. The operator Ak(t) is bounded by (25). Changing the order of summation by the Tonelli’s theorem, for every g∈Xk,
(29)∥Bk(t)g∥1=∫ℝ3‍(∑n=1∞‍n∑m=n+1k‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)|g(t,y,m)|dy∑n=1∞‍)dx=∫ℝ3(∑n=1∞‍n∑m=n+1k‍a(t,y,m)b(t,y,n,m)|g(t,y,m)|)dy‍=∫ℝ3∑m=2ka(t,y,m)|g(t,y,m)|(∑n=1∞‍nb(t,y,n,m))dy‍=∫ℝ3∑m=2k‍‍a(t,y,m)|g(t,y,m)|(∑n=1m-1‍nb(t,y,n,m))dy=∫ℝ3‍∑m=2k‍ma(t,y,m)|g(t,y,m)|dy=∫ℝ3‍∑m=1k‍ma(t,y,m)|g(t,y,m)|dy≤∫ℝ3‍∑m=1∞‍ma(t,y,m)|g(t,y,m)|dy=∥Ak(t)g∥1=Θt,k∥g∥1<∞,
where we have used (2) and (5), respectively. Then Bk(t) is also bounded. Hence Qk(t) generates a uniformly continuous semigroup. We denote this semigroup by (Gt,k(s))s≥0. Clearly, Ak(t) generates a positive semigroup of contractions and Bk(t) is a positive operator. Moreover, the above calculations also imply that D(Bk(t))⊃D(Ak(t)) and
(30)∑n=1∞‍∫ℝ3‍n(Ak(t)g(t,x,n)+Bk(t)g(t,x,n))dx=∑n=1k‍∫ℝ3‍n(Ak(t)g(t,x,n)+Bk(t)g(t,x,n))dx≤0
for all g∈(D(Ak(t)))+ with
(31)D(Ak(t))={g∈𝒳1;-atg∈Xk},D(Bk(t))={g∈𝒳1;Bk(t)g∈Xk}.
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 Qk(t) becomes independent of time [3, Lemma 2.1] and Kato’s theorem is immediately applicable. Therefore there is an extension Q˘k(t) of Qk(t) which generates a substochastic semigroup. Because at(x,n) is bounded in ℝ3×(0,k), this substochastic semigroup is conservative and it follows that Q˘k(t)=Q¯k(t), where Q¯k(t) is the closure of Qk(t). Since Qk(t) generates a uniformly (and hence strongly) continuous semigroup, Qk(t) is a closed operator. Therefore we have that Q˘k(t)=Qk(t); consequently, the uniformly continuous semigroup (Gt,k(s))s≥0 is a positive strongly continuous semigroup of contractions and furthermore (Gt,k(s))s≥0 is honest.
The proof of (27) is clear since we can use the usual power series definition to define Gt,k(s)=exp(sQk(t)). By induction, (Qk(t))J=(Q(t)qk)J=(Q(t))Jqk for J=1,2,…, from which the exponential formula yields (27).
To prove (28), we have that B(t)qkg=qkB(t)qkg on B(t)¯(0ℝ3,k)×[0,k] since for k≥n≥0, B(t)qkg(t,x,n) is given by (21) and B(t)qkg(t,x,n)=0 for k<n. Moreover, it is obvious that A(t)qkg=qkA(t)qkg; hence we have also
(32)(A(t)+B(t))qk=A(t)qk+B(t)qk=qkA(t)qk+qkB(t)qk=Ak(t)+Bk(t)=Qk(t).
Next, by qkqr=qrqk=qk we have
(33)qkQr(t)qk=qkqrQ(t)qrqk=qkQ(t)qk=Qk(t);
if we assume, by induction, that qk(Qr(t))j-1qk=(Qk(t))j-1, then
(34)qk(Qr(t))jqk=qk(Qr(t))j-1Qr(t)qk=qk(Qr(t))j-1qrQ(t)qrqk=qk(Qr(t))j-1qrqkQ(t)qk=qk(Qr(t))j-1qkQk(t)=(Qk(t))j.
Now using (27) and the the fact that Qr(t) is a bounded operator, the semigroup generated by Qr(t) is expressed by
(35)qkGt,r(s)qk=∑n=0∞‍sjqk(Qr(t))jqkn!=∑n=0∞‍sj(Qk(t))jj!=Gt,r(s),
which concludes the lemma.Next we assume thata(t,x,n) satisfies the Lipschitz condition
(36)|a(t,x,n)-a(σ,x,n)|≤|t-σ|Λ(x,n),t,σ∈ℑ,
where Λ(x,n)≥0 together with Λ(x,n)≤Θk; for all k≥n>0 we state the following lemma.Lemma 3.
The functiont→Qk(t) is continuous in the uniform operator topology for each k fixed in ℕ.Proof.
Using Fubini’s theorem and assumption (36) yields
(37)∥Qk(t)g-Qk(σ)g∥1≤kΘk|t-σ|∥g∥1,∀g∈𝒳1,
and the result follows.Making use of (14) and Lemma 2, there is a forward propagator, say {Uk(t,τ)}0≤τ≤t≤T defined in Xk⊂𝒳1 which is associated with the evolution semigroup (Gt,k(s))s≥0, t∈ℑ. The propagator {Uk(t,τ)}0≤τ≤t≤T shares certain properties with the family of semigroups (Gt,k(s))s≥0, t∈ℑ, as stated in the following theorem and proved in [3, Theorem 4.1].Theorem 4 (see [11]).
For eachk∈ℕ, the forward propagator {Uk(t,τ)}0≤τ≤t≤T generated by the family of infinitesimal generators {Qk(t)}t∈ℑ have the following properties:(1)
Uk(t,τ) is positive;(2)
∥Uk(t,τ)g∥1=∥g∥1, for all g∈𝒳1;(3)
∑n=0∞‍n[Uk(t,τ)g](t,x,n)=∑n=0∞‍ng(t,x,n), for all g∈𝒳1, x∈ℝ3;(4)
(∂/∂t)Uk(t,τ)=Qk(t)Uk(t,τ), 0≤τ≤t≤T;(5)
(∂/∂τ)Uk(t,τ)=-UkQk(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)=pτ0∈Xk. The solution is given by p(t)=Uk(t,τ0)pτ0  (0≤τ0≤t≤T).Proof.
This theorem is an immediate consequence of Lemmas2 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 {Qk(t)}t∈ℑ of C0-semigroups on Xk is stable with stability constants 1 and 0. Moreover we have from Theorem 4(38)∥Uk(t,τ)∥1≤1,for0≤τ≤t≤T.


## 3. Cauchy Problem for the Transport Model inℝ3×ℕ

We consider the Cauchy problem for the transport equation(39)∂∂tp(t,x,n)=-div(ω(x,n)p(t,x,n)),p(0,x,n)=pon(x),n=1,2,3,…,
in the space 𝒳1.We consider the functionω: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
(40)[D~p](t,x,n):=-div(ω(x,n)p(t,x,n))=(∇·ω(x,n))p(t,x,n)+ω(x,n)·(∇p(t,x,n)).
We assume that  (H1)
ω is divergence-free;(H2)
ω globally Lipschitz continuous; then divω(x,n):=∇·ω(x,n)=0 and (40) becomes
(41)[𝒟~p](t,x,n):=ω(x,n)·(∇p(t,x,n)).
We define in 𝒳1 the operators (𝒟,D(𝒟)) as
(42)[𝒟p](t,x,n)=[𝒟~p](t,x,n),with[𝒟~p](t,x,n)representedby(41)D(𝒟):={p∈𝒳1,ω·(∇p)𝒳1},n∈ℕ.Remark 7.
With the assumptions (H1)-(H2), it is proven [2] that 𝒟 is the generator of a strongly continuous stochastic semigroup, say (G𝒟(t))t≥0 and ∥G𝒟(t)∥≤1=1e0t. 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 anyg∈𝒳1,
(43)limλ→∞λR(λ,𝒟)g=g;(ii)
𝒟R(λ,𝒟) are bounded operators and for any g∈D(𝒟),
(44)limλ→∞λ𝒟R(λ,𝒟)g=𝒟g.


## 4. Perturbed Approximated Problem

Let𝒟λ=λ𝒟R(λ,𝒟); Yosida [13] was the first to use the bounded operators 𝒟λ to approximate the unbounded operator 𝒟, for which we can define semigroups via the exponential formula
(45)et𝒟=I+t𝒟1!+t2𝒟22!+⋯.
We exploit this idea to analyze the following approximated problem associated with (3):
(46)∂∂tp(t,x,n)=Qk(t)p(t,x,n)+[𝒟λp](t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3.Lemma 8.
Let eachk be fixed in ℕ and any λ>0. Under the assumptions (5), (25), (26), (36), (H1) and (H2), the operator (Tk,λ(t)=Qk(t)+𝒟λ,D(Qk(t))), (0≤t≤T) appearing in approximated Cauchy problem (46) is a stable generator, with the stability constants 1 and ∥𝒟λ∥1, of a forward propagator {𝕌k,λ(t,τ)}0≤τ≤t≤T associated with an evolution semigroup (𝔾t,k,λ(s))s≥0, positive, conserving the norm ∥·∥1 and given by
(47)𝔾t,k,λ(s)g=Gt,k(s)g+∑i=1∞‍Gλi(s)g,
where
(48)Gλi(s)g=∫0s‍Gt,k(s-v)g𝒟λGλi-1(v)gdv,g∈Xk
with (Gt,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)=pτ0∈Xk. The solution is given by p(t)=𝕌k,λ(t,τ0)pτ0  (0≤τ0≤t≤T).Proof.
Let us fixk∈ℕ and λ>0. The fact that Tk,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(Qk(t))⊃(0,∞). If ν>∥𝒟λ∥1, it is obvious that ν∈ρ(Tk,λ(t)) (bounded perturbation) and the resolvent satisfies
(49)R(ν,Tk,λ(t))=∑i=1∞‍R(ν,Qk(t))[𝒟λR(ν,Qk(t))]i.
Henceforth, for any finite sequence 0≤t1≤t2≤⋯≤tr≤T, r=1,2,…, we have
(50)∏n=1r‍R(ν,Tk,λ(tn))=∏n=1r‍[∑i=1∞‍R(ν,Qk(tn))[𝒟λR(ν,Qk(tn))]-1].
The development of the right-hand side of (50) yields a series with the general term in the form
(51)R(ν,Qk(tm))[𝒟λR(ν,Qk(tm))]im⋯R(v,Qk(t1))[𝒟λR(ν,Qk(t1))]i1,
where in≥0. If ∑n=1rin=i; then using the stability of the family Qk(t)0≤τ0≤t≤T, we estimate (51) by ∥𝒟λ∥1ν-i-r. Therefore
(52)∥∏n=1r‍R(ν,Tk,λ(tn))∥1≤ν-r∑i=1∞‍(ri)(∥𝒟λ∥1ν-1)-i=[(ν-∥𝒟λ∥1)]-1,
where (ri) is the number of terms in which ∑n=1rin=i in this series. To prove (47), we use the Duhamel equation [14]
(53)𝔾t,k,λ(s)g=Gt,k(s)g+∫0s‍Gt,k(s-v)g𝒟λ𝔾t,k,λ(v)gdv,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiig∈Xk
whose iteration leads to Dyson-Phillips series, see [15], given by
(54)𝔾t,k,λ(s)=∑i=0∞‍Gλi(s),withGλ0(s)g=Gt,k(s)g
and (47) follows.
The second part of the theorem follows from Theorem5 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
(55)0=∫ℝ3×ℕ𝒟pdμdm1,=∫ℝ3∑n=1∞‍n𝒟p(t,x,n)dx,
for all p∈D(𝒟)t≥0, which leads to
(56)ddtU(t)=ddt(∑n=1∞‍n∫ℝ3p(t,x,n)dx)=∑n=1∞‍n∫ℝ3∂tp(t,x,n)dx=∑n=1∞‍n∫ℝ3𝒟p(t,x,n)dx=0,
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
(57)∂∂tp(t,x,n)=∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)p(t,y,m)dy-a(t,x,n)p(t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3
has a solution p defined on the product set [0,∞)×ℝ×ℕ whenever pτ0∈𝒳1 and given by
(58)p(t,x,n)=[U(t,τ0)pτ0](x,n),n∈ℕ,0≤τ0<t,
where
(59)U(t,τ)g=limk→∞Uk(t,τ)g
for all g∈𝒳1 with {Uk(t,τ)}0≤τ<t≤T defined in Theorem 4. In the same way, if (Gt(s))s≥0 is the evolution semigroup associated with U(t,τ), (see (14)) then
(60)Gt(s)g=limk→∞Gt,k(s)g
for all g∈𝒳1 with {Gt,k(s)}s≥0 defined in Lemma 2. Because 𝒟λ is bounded, it follows that the perturbed Cauchy problem
(61)∂∂tp(t,x,n)=𝒟λp(t,x,n)+∑m=n+1∞‍∫ℝ3‍a(t,y,m)b(t,y,n,m)×h(x,n,m,y)p(t,y,m)dy-a(t,x,n)p(t,x,n),p(τ0,x,n)=pτ0(x,n),0≤τ0<t≤T,n=1,2,3,…,x∈ℝ3
also has a solution p defined on the product set [0,∞)×ℝ×ℕ whenever pτ0∈𝒳1. We can state the following.Lemma 9.
Let fixλ>0 and consider the families (𝔾t,k,λ(s))s≥0 and {𝕌k,λ(t,τ)}0≤τ≤t≤T defined in Lemma 8. The family (𝔾t,λ(s))s≥0 defined by
(62)𝔾t,λ(s)g=limk→∞𝔾t,k,λ(s)g,g∈𝒳1
exists for all 0≤t≤T and (𝔾t,λ(s))s≥0 forms a positive, C0-semigroup on 𝒳1 conserving the norm ∥·∥1.
In the same way the family{𝕌λ(t,τ)}0≤τ≤t≤T defined by
(63)𝕌λ(t,τ)g=limk→∞𝕌k,λ(t,τ)g,g∈𝒳1
exists for all 0≤τ<t≤T and {𝕌λ(t,τ)}0≤τ≤t≤T forms a forward propagator on 𝒳1 conserving the norm ∥·∥1.Proof.
Letg∈𝒳1; from (27) and (28) we have
(64)𝔾t,k,λ(s)qrg=𝔾t,r,λ(s)g+qrg-g,wherek≥r.
Then
(65)∥𝔾t,k,λ(s)g-𝔾t,r,λ(s)g∥1≤∥𝔾t,k,λ(s)g-𝔾t,k,λ(s)qrg∥1+∥g-qrg∥1=2∥g-qrg∥1⟶0,hhhhhhhhhhhhhhhask,r⟶∞.
Hence, the family (𝔾t,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 𝔾t,λ(s):𝒳1→𝒳1. Furthermore
(66)∥𝔾t,λ(s)g∥1=∥limk→∞𝔾t,k,λ(s)g∥1=limk→∞∥𝔾t,k,λ(s)g∥1=limk→∞∥g∥1=∥g∥1,
where we have used the continuity of ∥·∥1 on 𝒳1 and the fact that {𝔾t,k,λ(s)} conserves the norm ∥·∥1. Thus {𝔾t,λ(s)}s≥0 conserves the norm ∥·∥1:
(
𝔾
t
,
k
,
λ
(
s
)
)
s
≥
0 is a semigroup, in fact
(a)(67)𝔾t,λ(s+v)g=limk→∞𝔾t,k,λ(s+v)g=limk→∞𝔾t,k,λ(s)limk→∞𝔾t,k,λ(v)g.
In the same way as previously, we show, using the definition of (𝔾t,λ(s))s≥0, that
(68)∥𝔾t,k,λ(s)𝔾t,k,λ(v)g-𝔾t,λ(s)𝔾t,λ(v)g∥1⟶0,hhhhhhhhhhhhhhhhhhhhhhhhhask,r⟶∞.
Hence
(69)𝔾t,λ(s)𝔾t,λ(v)g⟵𝔾t,k,λ(s)𝔾t,k,λ(v)g=𝔾t,k,λ(s+v)g⟶𝔾t,λ(s+v)ghhhhhhhhhhhhhhhhhhhhask⟶∞.
This convergence, being strong in 𝒳1 finally yields 𝔾t,λ(s)𝔾t,λ(v)=𝔾t,λ(s+v):
(b)𝔾t,λ(0)=I since
(70)𝔾t,λ(0)g=limk→∞𝔾t,k,λ(0)g=limk→∞g=g
for all g∈𝒳1.
(c) By a similar way we show thatlims→0+𝔾t,k,λg=g since the limits (62) exist uniformly in s.
(d){𝔾t,λ(s)} is positive since from the definition of 𝔾t,λ(s) and [16, Corollary 5.11], there exists a subsequence {𝔾t,kδ,λ(s)g}  (δ=1,2,…) so that
(71)[𝔾t,kδ,λ(s)g](x,n)⟶[𝔾t,λ(s)g](x,n),asδ⟶∞fora.e.(x.n)∈ℝ3×ℕ.
Because {𝔾t,kδ,λ(s)g} is positive then if g>0, this pointwise limit gives
(72)[𝔾t,λ(s)g](x,n)≥0fora.e.(x.n)∈ℝ3×ℕ,
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.
Letλ>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 (Tλ(t)=Q(t)+𝒟λ,D(Q(t))), (0≤t≤T) defining the Cauchy problem (61). Furthermore the solution given, for all initial data p(τ0)=pτ0∈𝒳1, by
(73)p(t,x,n)=[𝕌λ(t,τ0)pτ0](x,n),n∈ℕ,0≤τ0<t,
satisfies the perturbed Cauchy problem (61).


## 5. Existence Results: Discussion and Concluding Remark

Now what happens if we tendλ→∞!. 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.



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*Source: 484391-2013-12-30.xml*