# Global Solvability of a Continuous Model for Nonlocal Fragmentation Dynamics in a Moving Medium

Authors: S. C. Oukouomi Noutchie; E. F. Doungmo Goufo
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Existence of global solutions to continuous nonlocal convection-fragmentation equations is investigated in
spaces of distributions with finite higher moments. Under the assumption that the velocity field is divergence-free, we make use of the method of characteristics and Friedrichs's lemma  (Mizohata, 1973) to show that the transport operator
generates a stochastic dynamical system. This allows for the use of substochastic methods and Kato-Voigt
perturbation theorem  (Banasiak and Arlotti, 2006)  to ensure that the combined transport-fragmentation operator is the infinitesimal
generator of a strongly continuous semigroup. In particular, we show that the solution represented by this
semigroup is conservative.


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## Body

## 1. Motivation and Introduction

The process of fragmentation of clusters occurs in many branches of natural sciences ranging from physics, through chemistry, engineering, biology, to ecology and in numerous domains of applied sciences, such as the depolymerization, the rock fractures, and the breakage of droplets. The fragmentation rate can be size and position dependent, and new particles resulting from the fragmentation are spatially distributed across the space. Fragmentation equations, combined with transport terms, have been used to describe a wide range of phenomena. For instance, in ecology or aquaculture, we have phytoplankton population in flowing water. In chemical engineering, we have applications describing polymerization, polymer degradation, and solid drugs breakup in organisms or in solutions. We also have external processes such as oxidation, melting, or dissolution, which cause the exposed surface of particles to recede, resulting in the loss of mass of the system. Simultaneously, they widen the surface pores of the particle, causing the loss of connectivity and thus fragmentation, as the pores join each other (see [1–4] and references therein). Various types of pure fragmentation equations have been comprehensively analyzed in numerous works (see, e.g., [5–9]). Conservative and nonconservative regimes for pure fragmentation equations have been thoroughly investigated, and, in particular, the breach of the mass conservation law (called shattering) has been attributed to a phase transition creating a dust of “zero-size” particles with nonzero mass, which are beyond the model’s resolution. But fragmentation and transport processes combined in the same model are still barely touched in the domain of mathematical and abstract analysis. Kinetic-type models with diffusion were globally investigated in [5] and later extended in [10], where the author showed that the diffusive part does not affect the breach of the conservation laws, and, very recently, in [11], the author investigated equations describing fragmentation and coagulation processes with growth or decay and proved an analogous result.In this paper, we present and analyze a special and noncommon type of transport process. In social grouping population, if we define a spatial dynamical system in which locally group-size distribution can be estimated, but in which we also allow immigration and emigration from adjacent areas with different distributions, we obtain the general model consisting of transport, direction changing, and fragmentation and coagulation processes describing the dynamics a population of, for example, phytoplankton, which is a spatially explicit group-size distribution model as presented in [12]. We analyze, in this work, the model consisting of transport and fragmentation processes, hoping that it will bring a significant contribution to the analysis of the full problem (with transport, direction changing, and fragmentation and coagulation processes) which remains an open problem.


## 2. Well Posedness of the Transport Problem with Fragmentation

We consider the following Cauchy problem [12]:
(1)∂∂tp(t,x,m)=-div(ω(x,m)p(t,x,m))-a(x,m)p(t,x,m)+∫m∞b(x,s,m)a(x,s)p(t,x,s)ds,p(0,x,m)=po(x,m),a.e.(x,m)∈ℝ3×ℝ+,
where, in terms of the mass size m and the position x, the state of the system is characterized at any moment t by the particle-mass-position distribution p=p(t,x,m) (p is also called the density or concentration of particles), with p:ℝ+×ℝ3×ℝ+→ℝ+. The three-dimensional vector ω=ω(x,m) represents the velocity of the transport and is supposed to be a known quantity depending on m and x; a(x,m) is the average fragmentation rate; that is, it describes the ability of aggregates of size m and position x to break into smaller particles. Once an aggregate of mass s and position x breaks, the expected number of daughter particles of size m is the nonnegative measurable function b(x,s,m) defined on ℝ3×ℝ×ℝ. The space variable x is supposed to vary in the whole of ℝ3. The function po(x,m) represents the density of groups of size m at position x at the beginning (t=0).


### 2.1. Fragmentation Equation

Let us introduce necessary assumptions that will be useful in our analysis. Since a group of sizem≤s cannot split to form a group of size s, the function b(x,s,m) has its support in the set
(2)ℝ3×{(s,m)∈ℝ+×ℝ+:m<s}.
After the fragmentation of a mass s particle, the sum of masses of all daughter particles should again be s; hence it follows that, for any s>0, x∈ℝ3(3)∫0smb(x,s,m)dm=s.
Because the space variable x varies in the whole of ℝ3 (unbounded) and since the total number of individuals in a population is not modified by interactions among groups, the following conservation law is supposed to be satisfied:
(4)ddt𝒩(t)=0,
where 𝒩(t)=∫ℝ3∫0∞p(t,x,m)mdmdx is the total number of individuals in the space (or total mass of the ensemble). Since p=p(t,x,m) is the density of groups of size m at the position x and time t and that mass is expected to be a conserved quantity, the most appropriate Banach space to work in is the space
(5)𝒳1:=L1(ℝ3×ℝ+,mdmdx).
But because uniqueness of solutions of (1) proved to be a more difficult problem [11], we restrict our analysis to a smaller class of functions, so we introduce the following class of Banach spaces (of distributions with finite higher moments):(6)𝒳r:=L1(ℝ3×ℝ+,mrdmdx),r≥1,
which coincides with 𝒳1 for r=1 and is endowed with the norm ∥·∥r. We assume that po∈𝒳r, and, for each t≥0, the function (x,m)→p(x,m)=p(t,x,m) is from the space 𝒳r with r≥1. When any subspace S⊆𝒳r, we will denote by S+ the subset of S defined as S+={g∈S;g(x,m)≥0,m∈ℝ+,x∈ℝ3}. Note that any g∈(𝒳r)+ will possess moments
(7)Mq(t):=∫0∞mqg(t,x,m)dm
of all orders q∈[0,r]. In 𝒳r, we define from the expressions on the right-hand side of (1) the operators A and B by
(8)[Ag](x,m):=a(x,m)g(x,m),D(A):={g∈𝒳r:ag∈𝒳r},(9)[Bg](x,m):=∫m∞b(x,s,m)a(x,s)p(x,s)ds,D(B):=D(A).Lemma 1.
(
A
+
B
,
D
(
A
)
) is a well-defined operator.Proof.
To prove thatB is well defined on D(A) as stated in (9), we use the condition (3) to show that
(10)sr-∫0smrb(x,s,m)dm=sr-∫0smr-1mb(x,s,m)dm≥sr-sr-1∫0smb(x,s,m)dm=0.
Hence
(11)∫0smrb(x,s,m)dm≤sr
for r≥1,m>0. Note that the equality holds for r=1. For every p∈D(A)+, changing the order of integration by the Fubini theorem, we have
(12)∥Bp∥r=∫ℝ3∫0∞[Bp](x,m)mrdmdx=∫ℝ3∫0∞(∫m∞b(x,s,m)a(x,s)p(x,s)mrds)dmdx=∫ℝ3∫0∞(∫0sb(x,s,m)a(x,s)p(x,s)mrdm)dsdx≤∫ℝ3∫0∞a(x,s)p(x,s)srdsdx=∥Ap∥r<∞,
where we have used inequality (11). The result follows from the fact that any arbitrary element p of D(A) can be written in the form p=p+-p-, where p+,p-∈D(A)+. Then ∥Bp∥r≤∥Ap∥r, for all p∈D(A), so that we can take D(B):=D(A), and (A+B,D(A)) is well defined.


### 2.2. Cauchy Problem for the Transport Operator inΛ=ℝ3×ℝ+

Λ is endowed with the Lebesgue measure dμ=dμm,x=dmdx. Our primary objective in this section is to analyze the solvability of the transport problem
(13)∂∂tp(t,x,m)=-div(ω(x,m)p(t,x,m)),p(0,x,m)=po(x,m),m∈ℝ+,x∈ℝ3
in the space 𝒳r.Furthermore, to simplify the notation we putv=(x,m)∈Λ. We consider the function ω:Λ→ℝ3 and 𝒟~ the expression appearing on the right-hand side of (13). Then
(14)𝒟~[p(t,v)]:=-div(ω(v)p(t,v))=(∇·ω(v))p(t,v)+ω(v)·(∇p(t,v)).
We assume that ω is divergence-free and globally Lipschitz continuous. Then divω(v):=∇·ω(v)=0, and (14) becomes
(15)𝒟~[p(t,v)]:=ω(v)·(∇p(t,v)).
For v∈Λ and t∈ℝ, the initial value problem
(16)d𝔯ds=ω(𝔯),s∈ℝ,𝔯(t)=v
has one and only one solution 𝔯(s) taking values in Λ. Thus we can consider the function ϕ:Λ×ℝ2→Λ defined by the condition that, for (v,t)∈Λ×ℝ,
(17)s⟶ϕ(v,t,s),s∈ℝ,
is the only solution of the Cauchy problem (16). The integral curves given by the ϕ-parameter family (𝔯)ϕ (with 𝔯(s)=ϕ(v,t,s),s∈ℝ, the only solution of (16)) are called the characteristics of 𝒟~. The function ϕ possesses many desirable properties [13–15] that will be relevant for studying the transport operator in 𝒳r. Some of them are listed in [5, Proposition 10.1]. Now we can properly study the transport operator 𝒟. Using the above proposition in our application, we can take
(18)𝒟p=𝒟~p,with𝒟~prepresentedby(15),D(𝒟):={p∈𝒳r,𝒟p∈𝒳r}.Note that𝒟p is understood in the sense of distribution. Precisely speaking, if we take C01(Λ) as the set of the test functions, each p∈D(𝒟) if and only if p∈L1(Λ), and there exists g∈𝒳r such that
(19)∫Λ‍ξgdμ=∫Λ‍p∂·(ξω)dμ=∫Λ‍pω·∂ξdμ,
for all ξ∈C01(Λ), where
(20)ω·∂ξ(v):=∑j=13‍ωj∂jξ(v)
with ωj=ωj(v), the jth component of the velocity ω(v). The middle term in (19) exists as ω is globally Lipschitz continuous, and the last equality follows as ω is divergence-free. If this is the case, we define 𝒟p=g.Now we can show that the operator𝒟 is the generator of a stochastic semigroup on 𝒳r.Theorem 2.
If the functionω is globally Lipschitz continuous and divergence-free, then the operator (D(𝒟),𝒟) defined by (18) is the generator of a strongly continuous stochastic semigroup (G𝒟(t))t≥0, given by
(21)[G𝒟(t)p](v)=p(ϕ(v,t,0))
for any p∈𝒳r and t>0.Proof.
Let(Z0(t))t≥0 be the family defined by the right-hand side of the relation (21). The proof of the theorem will follow three steps.
(i) First we show that(Z0(t))t≥0 is a strongly continuous semigroup of bounded linear operators. We need some properties of ϕ as listed in [5] and given as follows. The function ϕ has the following properties:(p1)
ϕ(v,t,t)=v for all v∈Λ,t∈ℝ;(p2)
ϕ(ϕ(v,t,s),s,τ)=ϕ(v,t,τ) for all v∈Λ,t,s, and τ∈ℝ;(p3)
ϕ(v,t,s)=ϕ(v,t-s,0)=ϕ(v,0,s-t) for all v∈Λ,t,s∈ℝ;(p4)
|ϕ(v,t,s)-ϕ(y,t,s)|≤eK|t-s||v-y| for all v,y∈Λ,t,s∈ℝ;(p5)
functionΛ×ℝ×ℝ∋(v,t,s)→ϕ(v,t,s) is continuous;(p6)
the transformation𝒯 defined by t=t,s=s, and y=ϕ(v,t,s) is a topological homeomorphism which is bimeasurable, and its inverse 𝒯-1 is represented by t=t,s=s, and v=ϕ(y,s,t);(p7)
for allt,s∈ℝ the transformation of Λ onto itself defined by y=ϕ(v,t,s) is measure preserving.
Then by the properties (p6) and (p7), we see that, for any p, the composition (v,t)→p(ϕ(v,t,0)), in (21), is a measurable function satisfying the equality
(22)∥Z0(t)p∥r=∥p∥r.
Hence the family (Z0(t))t≥0 is of bounded linear operators from 𝒳r→𝒳r. Then we can easily verify the following relations:  (ia)
Z0(0)=I;(ib)
Z0(t+s)=Z0(t)Z0(s), for all t,s∈ℝ;(ic)
limt→0+∥Z0(t)p-p∥r=0, for each p∈𝒳r.
In fact, (ia) and (ib) follow immediately from the properties (p1) and (p2). To prove (ic), we can follow the argument of Example 3.10 in [5]. Thus, it is enough to show (ic) for p∈C0∞(Λ). For such ps, we have limt→0+(Z0(t)p)(v)=p(v) for all v∈Λ. Furthermore, if |p(v)|≤M for all v∈Λ, then |(Z0(t)p)(v)|≤M for all v∈Λ, and, because the support of Z0(t)p is bounded, the Lebesgue dominated convergence theorem shows that (ic) is satisfied. Thus (Z0(t))t≥0 is a C0-semigroup.
(ii) Secondly, we prove that the generatorT0 of (Z0(t))t≥0 is an extension of 𝒟.
Let𝒴 be the set of real-valued functions which are defined on Λ, are Lipschitz continuous, and compactly supported. Obviously 𝒴⊂D(𝒟) because if p∈𝒴, then the first-order partial derivatives of p are measurable, bounded, and compactly supported and thus, multiplied by Lipschitz continuous functions of ω, belong to L1(Λ,dμ). For any fixed p∈𝒴,  we denote by ϑ the real-valued function defined on Λ×R+ by(23)ϑ(v,t)=(Z0(t)p)(v).
From the previous considerations and properties (p3)–(p5) there exists a measurable subset E of Λ×R+, with μ(Λ×R+∖E)=0, such that at each point (v,t)∈E the function ϑ has measurable first-order partial derivatives. In particular,(24)∂ϑ∂t(v,t)=(Z0(t)𝒟p)(v),(v,t)∈E,
and, therefore, if we let λp:=esssup(v)∈Λ|𝒟p|, then
(25)|∂tϑ(v,t)|≤λp
for any (v,t)∈E. From this and from part (i) of the proof it follows that, for every h>0,
(26)∥h-1(Z0(h)p-p)-𝒟p∥r=∥h-1∫0h(Z0(s)-I)𝒟pds∥r⟶0
as h→0+. This proves that 𝒴⊂D(T0) and that T0p=𝒟p, for all p∈𝒴. Next we prove that 𝒴 is a core of 𝒟, that is, that (𝒟,D(𝒟)) is the closure of (𝒟,𝒴). Let ϖɛ,ɛ>0, be a mollifier (see Example 2.1 in [5]), and, for p, let ϖɛ*p be the mollification of p. We use the Friedrichs lemma, [16, pp. 313–315], or [17, Lemma 1.2.5], which states that there is C>0, independent of ɛ, such that for any Lr function p,1≤r<∞, we have
(27)∥𝒟(ϖɛ*p)-ϖɛ*𝒟p∥r≤C∥p∥r,(28)limɛ→0+(∥ϖɛ*p-p∥r+∥𝒟(ϖɛ*p)-𝒟p∥r)=0.
Estimates of Equation (2.9) in [5] and the above relation (27) imply(29)∥𝒟(ϖɛ*p)∥r≤C∥p∥r+∥𝒟p∥r
which shows that the mollification p→ϖɛ*p is a continuous operator in D(𝒟) (equipped with the graph norm) uniformly bounded with respect to ɛ. Next we observe that the subset of D(𝒟) consisting of compactly supported functions is dense in D(𝒟) with the graph norm. Indeed, let p∈D(𝒟). Because both p,𝒟p∈𝒳r, the absolute continuity of the Lebesgue integral implies that for any given δ>0 there exists a compact subset Ω′ of Λ such that
(30)∫Λ∖Ω′‍(|p|+|𝒟p|)dμ<δ.
For this Ω′ we choose ψ∈C0∞(Λ) satisfying 0≤ψ(v)≤1 for all v∈Λ, and ψ(v)=1 for all v∈Ω′. Now it is easy to see that ψp∈D(𝒟) and has a compact support. Moreover,
(31)∥ψp-p∥r≤2∫Λ∖Ω′‍|p|dμ,∥𝒟(ψp)-𝒟p∥r≤2∫Λ∖Ω′‍|𝒟p|dμ+L∫Λ∖Ω′‍|p|dμ,
whereL=sup|𝒟ψ| can be made independent of Ω′ due to the fact that Λ is the whole space.
Letp∈D(𝒟) be compactly supported. From Example 2.1 in [5] we know that ϖɛ*p is infinitely differentiable and compactly supported and thus belongs to 𝒴. Equation (28) yields that ϖɛ*p→p as ɛ→0+ in the graph norm of D(𝒟). Because we have shown above that compactly supported functions from D(𝒟) are dense in D(𝒟), we see that (𝒟,D(𝒟)) is the closure of (𝒟,𝒴), and, because T0 is a closed extension of (𝒟,𝒴), we obtain 𝒟⊂T0.
(iii) Lastly we recognize thatD(T0)⊂D(𝒟) so that the operators T0 and 𝒟 coincide, and (G𝒟(t))t≥0=(Z0(t))t≥0. Suppose p∈D(T0). Then for any fixed λ>0 there exists a unique g∈𝒳r such that p=(λI-T0)-1g. For any ψ∈C01(Λ) we have, by (19),
(32)∫Λ𝒟pψdμ=∫Λp(v)(ω·∂ψ)(v)dμv=∫Λ(∫0∞e-λtg(ϕ(v,t,0))dt)(ω·∂ψ)(v)dμv=∫0∞(∫Λe-λtg(ϕ(v,t,0))(ω·∂ψ)(v)dμv)dt=∫0∞(∫Λe-λtg(y)(ω·∂ψ)(ϕ(y,0,t))(v)dμy)dt=∫Λ(∫0∞e-λtddtψ(ϕ(y,0,t))dt)g(y)dμy=∫Λ(λ∫Λ‍(∫0∞‍e-λtψ(ϕ(y,0,t))dt)e-λtψ(ϕ(y,0,t))|0∞g(y)dy+λ∫Λ(∫0∞e-λtψ(ϕ(y,0,t))dt)∫Λ(λ∫Λ‍(∫0∞‍e-λtψ(ϕ(y,0,t))dt)e-λtψ(ϕ(y,0,t))|0∞g(y)dy)g(y)dμy=-∫Λg(y)ψ(y)dμy+λ∫Λ(∫0∞e-λtg(ϕ(v,0,t))dt)ψ(v)dμv=-∫Λ(g-λp)ψdμ.
This implies thatp∈D(𝒟). Hence T0⊂𝒟, and 𝒟p=T0p.Remark 3 (conservativeness of the transport model).
Because the flow process does not modify the total number of individuals in the system, let us show that the model (13) is conservative in the space 𝒳r; that is, the law (4) is satisfied. We have proved that the semigroup generated by the operator 𝒟 is stochastic; then we have
(33)0=∫Λ𝒟pdμ,∀p∈D(𝒟),then0=∫ℝ3∫0∞mr𝒟p(t,x,m)dmdx,∀t≥0,r≥1.
Thus, ∫ℝ3∫0∞m𝒟p(t,x,m)dmdx=0,   for all t≥0 which leads to
(34)ddt𝒩(t)=ddt(∫ℝ3∫0∞mp(t,x,m)dmdx)=∫ℝ3∫0∞m∂tp(t,x,m)dmdx=∫ℝ3∫0∞m𝒟p(t,x,m)dmdx=0
and therefore proving the conservativeness of the transport model in (18).


## 2.1. Fragmentation Equation

Let us introduce necessary assumptions that will be useful in our analysis. Since a group of sizem≤s cannot split to form a group of size s, the function b(x,s,m) has its support in the set
(2)ℝ3×{(s,m)∈ℝ+×ℝ+:m<s}.
After the fragmentation of a mass s particle, the sum of masses of all daughter particles should again be s; hence it follows that, for any s>0, x∈ℝ3(3)∫0smb(x,s,m)dm=s.
Because the space variable x varies in the whole of ℝ3 (unbounded) and since the total number of individuals in a population is not modified by interactions among groups, the following conservation law is supposed to be satisfied:
(4)ddt𝒩(t)=0,
where 𝒩(t)=∫ℝ3∫0∞p(t,x,m)mdmdx is the total number of individuals in the space (or total mass of the ensemble). Since p=p(t,x,m) is the density of groups of size m at the position x and time t and that mass is expected to be a conserved quantity, the most appropriate Banach space to work in is the space
(5)𝒳1:=L1(ℝ3×ℝ+,mdmdx).
But because uniqueness of solutions of (1) proved to be a more difficult problem [11], we restrict our analysis to a smaller class of functions, so we introduce the following class of Banach spaces (of distributions with finite higher moments):(6)𝒳r:=L1(ℝ3×ℝ+,mrdmdx),r≥1,
which coincides with 𝒳1 for r=1 and is endowed with the norm ∥·∥r. We assume that po∈𝒳r, and, for each t≥0, the function (x,m)→p(x,m)=p(t,x,m) is from the space 𝒳r with r≥1. When any subspace S⊆𝒳r, we will denote by S+ the subset of S defined as S+={g∈S;g(x,m)≥0,m∈ℝ+,x∈ℝ3}. Note that any g∈(𝒳r)+ will possess moments
(7)Mq(t):=∫0∞mqg(t,x,m)dm
of all orders q∈[0,r]. In 𝒳r, we define from the expressions on the right-hand side of (1) the operators A and B by
(8)[Ag](x,m):=a(x,m)g(x,m),D(A):={g∈𝒳r:ag∈𝒳r},(9)[Bg](x,m):=∫m∞b(x,s,m)a(x,s)p(x,s)ds,D(B):=D(A).Lemma 1.
(
A
+
B
,
D
(
A
)
) is a well-defined operator.Proof.
To prove thatB is well defined on D(A) as stated in (9), we use the condition (3) to show that
(10)sr-∫0smrb(x,s,m)dm=sr-∫0smr-1mb(x,s,m)dm≥sr-sr-1∫0smb(x,s,m)dm=0.
Hence
(11)∫0smrb(x,s,m)dm≤sr
for r≥1,m>0. Note that the equality holds for r=1. For every p∈D(A)+, changing the order of integration by the Fubini theorem, we have
(12)∥Bp∥r=∫ℝ3∫0∞[Bp](x,m)mrdmdx=∫ℝ3∫0∞(∫m∞b(x,s,m)a(x,s)p(x,s)mrds)dmdx=∫ℝ3∫0∞(∫0sb(x,s,m)a(x,s)p(x,s)mrdm)dsdx≤∫ℝ3∫0∞a(x,s)p(x,s)srdsdx=∥Ap∥r<∞,
where we have used inequality (11). The result follows from the fact that any arbitrary element p of D(A) can be written in the form p=p+-p-, where p+,p-∈D(A)+. Then ∥Bp∥r≤∥Ap∥r, for all p∈D(A), so that we can take D(B):=D(A), and (A+B,D(A)) is well defined.


## 2.2. Cauchy Problem for the Transport Operator inΛ=ℝ3×ℝ+

Λ is endowed with the Lebesgue measure dμ=dμm,x=dmdx. Our primary objective in this section is to analyze the solvability of the transport problem
(13)∂∂tp(t,x,m)=-div(ω(x,m)p(t,x,m)),p(0,x,m)=po(x,m),m∈ℝ+,x∈ℝ3
in the space 𝒳r.Furthermore, to simplify the notation we putv=(x,m)∈Λ. We consider the function ω:Λ→ℝ3 and 𝒟~ the expression appearing on the right-hand side of (13). Then
(14)𝒟~[p(t,v)]:=-div(ω(v)p(t,v))=(∇·ω(v))p(t,v)+ω(v)·(∇p(t,v)).
We assume that ω is divergence-free and globally Lipschitz continuous. Then divω(v):=∇·ω(v)=0, and (14) becomes
(15)𝒟~[p(t,v)]:=ω(v)·(∇p(t,v)).
For v∈Λ and t∈ℝ, the initial value problem
(16)d𝔯ds=ω(𝔯),s∈ℝ,𝔯(t)=v
has one and only one solution 𝔯(s) taking values in Λ. Thus we can consider the function ϕ:Λ×ℝ2→Λ defined by the condition that, for (v,t)∈Λ×ℝ,
(17)s⟶ϕ(v,t,s),s∈ℝ,
is the only solution of the Cauchy problem (16). The integral curves given by the ϕ-parameter family (𝔯)ϕ (with 𝔯(s)=ϕ(v,t,s),s∈ℝ, the only solution of (16)) are called the characteristics of 𝒟~. The function ϕ possesses many desirable properties [13–15] that will be relevant for studying the transport operator in 𝒳r. Some of them are listed in [5, Proposition 10.1]. Now we can properly study the transport operator 𝒟. Using the above proposition in our application, we can take
(18)𝒟p=𝒟~p,with𝒟~prepresentedby(15),D(𝒟):={p∈𝒳r,𝒟p∈𝒳r}.Note that𝒟p is understood in the sense of distribution. Precisely speaking, if we take C01(Λ) as the set of the test functions, each p∈D(𝒟) if and only if p∈L1(Λ), and there exists g∈𝒳r such that
(19)∫Λ‍ξgdμ=∫Λ‍p∂·(ξω)dμ=∫Λ‍pω·∂ξdμ,
for all ξ∈C01(Λ), where
(20)ω·∂ξ(v):=∑j=13‍ωj∂jξ(v)
with ωj=ωj(v), the jth component of the velocity ω(v). The middle term in (19) exists as ω is globally Lipschitz continuous, and the last equality follows as ω is divergence-free. If this is the case, we define 𝒟p=g.Now we can show that the operator𝒟 is the generator of a stochastic semigroup on 𝒳r.Theorem 2.
If the functionω is globally Lipschitz continuous and divergence-free, then the operator (D(𝒟),𝒟) defined by (18) is the generator of a strongly continuous stochastic semigroup (G𝒟(t))t≥0, given by
(21)[G𝒟(t)p](v)=p(ϕ(v,t,0))
for any p∈𝒳r and t>0.Proof.
Let(Z0(t))t≥0 be the family defined by the right-hand side of the relation (21). The proof of the theorem will follow three steps.
(i) First we show that(Z0(t))t≥0 is a strongly continuous semigroup of bounded linear operators. We need some properties of ϕ as listed in [5] and given as follows. The function ϕ has the following properties:(p1)
ϕ(v,t,t)=v for all v∈Λ,t∈ℝ;(p2)
ϕ(ϕ(v,t,s),s,τ)=ϕ(v,t,τ) for all v∈Λ,t,s, and τ∈ℝ;(p3)
ϕ(v,t,s)=ϕ(v,t-s,0)=ϕ(v,0,s-t) for all v∈Λ,t,s∈ℝ;(p4)
|ϕ(v,t,s)-ϕ(y,t,s)|≤eK|t-s||v-y| for all v,y∈Λ,t,s∈ℝ;(p5)
functionΛ×ℝ×ℝ∋(v,t,s)→ϕ(v,t,s) is continuous;(p6)
the transformation𝒯 defined by t=t,s=s, and y=ϕ(v,t,s) is a topological homeomorphism which is bimeasurable, and its inverse 𝒯-1 is represented by t=t,s=s, and v=ϕ(y,s,t);(p7)
for allt,s∈ℝ the transformation of Λ onto itself defined by y=ϕ(v,t,s) is measure preserving.
Then by the properties (p6) and (p7), we see that, for any p, the composition (v,t)→p(ϕ(v,t,0)), in (21), is a measurable function satisfying the equality
(22)∥Z0(t)p∥r=∥p∥r.
Hence the family (Z0(t))t≥0 is of bounded linear operators from 𝒳r→𝒳r. Then we can easily verify the following relations:  (ia)
Z0(0)=I;(ib)
Z0(t+s)=Z0(t)Z0(s), for all t,s∈ℝ;(ic)
limt→0+∥Z0(t)p-p∥r=0, for each p∈𝒳r.
In fact, (ia) and (ib) follow immediately from the properties (p1) and (p2). To prove (ic), we can follow the argument of Example 3.10 in [5]. Thus, it is enough to show (ic) for p∈C0∞(Λ). For such ps, we have limt→0+(Z0(t)p)(v)=p(v) for all v∈Λ. Furthermore, if |p(v)|≤M for all v∈Λ, then |(Z0(t)p)(v)|≤M for all v∈Λ, and, because the support of Z0(t)p is bounded, the Lebesgue dominated convergence theorem shows that (ic) is satisfied. Thus (Z0(t))t≥0 is a C0-semigroup.
(ii) Secondly, we prove that the generatorT0 of (Z0(t))t≥0 is an extension of 𝒟.
Let𝒴 be the set of real-valued functions which are defined on Λ, are Lipschitz continuous, and compactly supported. Obviously 𝒴⊂D(𝒟) because if p∈𝒴, then the first-order partial derivatives of p are measurable, bounded, and compactly supported and thus, multiplied by Lipschitz continuous functions of ω, belong to L1(Λ,dμ). For any fixed p∈𝒴,  we denote by ϑ the real-valued function defined on Λ×R+ by(23)ϑ(v,t)=(Z0(t)p)(v).
From the previous considerations and properties (p3)–(p5) there exists a measurable subset E of Λ×R+, with μ(Λ×R+∖E)=0, such that at each point (v,t)∈E the function ϑ has measurable first-order partial derivatives. In particular,(24)∂ϑ∂t(v,t)=(Z0(t)𝒟p)(v),(v,t)∈E,
and, therefore, if we let λp:=esssup(v)∈Λ|𝒟p|, then
(25)|∂tϑ(v,t)|≤λp
for any (v,t)∈E. From this and from part (i) of the proof it follows that, for every h>0,
(26)∥h-1(Z0(h)p-p)-𝒟p∥r=∥h-1∫0h(Z0(s)-I)𝒟pds∥r⟶0
as h→0+. This proves that 𝒴⊂D(T0) and that T0p=𝒟p, for all p∈𝒴. Next we prove that 𝒴 is a core of 𝒟, that is, that (𝒟,D(𝒟)) is the closure of (𝒟,𝒴). Let ϖɛ,ɛ>0, be a mollifier (see Example 2.1 in [5]), and, for p, let ϖɛ*p be the mollification of p. We use the Friedrichs lemma, [16, pp. 313–315], or [17, Lemma 1.2.5], which states that there is C>0, independent of ɛ, such that for any Lr function p,1≤r<∞, we have
(27)∥𝒟(ϖɛ*p)-ϖɛ*𝒟p∥r≤C∥p∥r,(28)limɛ→0+(∥ϖɛ*p-p∥r+∥𝒟(ϖɛ*p)-𝒟p∥r)=0.
Estimates of Equation (2.9) in [5] and the above relation (27) imply(29)∥𝒟(ϖɛ*p)∥r≤C∥p∥r+∥𝒟p∥r
which shows that the mollification p→ϖɛ*p is a continuous operator in D(𝒟) (equipped with the graph norm) uniformly bounded with respect to ɛ. Next we observe that the subset of D(𝒟) consisting of compactly supported functions is dense in D(𝒟) with the graph norm. Indeed, let p∈D(𝒟). Because both p,𝒟p∈𝒳r, the absolute continuity of the Lebesgue integral implies that for any given δ>0 there exists a compact subset Ω′ of Λ such that
(30)∫Λ∖Ω′‍(|p|+|𝒟p|)dμ<δ.
For this Ω′ we choose ψ∈C0∞(Λ) satisfying 0≤ψ(v)≤1 for all v∈Λ, and ψ(v)=1 for all v∈Ω′. Now it is easy to see that ψp∈D(𝒟) and has a compact support. Moreover,
(31)∥ψp-p∥r≤2∫Λ∖Ω′‍|p|dμ,∥𝒟(ψp)-𝒟p∥r≤2∫Λ∖Ω′‍|𝒟p|dμ+L∫Λ∖Ω′‍|p|dμ,
whereL=sup|𝒟ψ| can be made independent of Ω′ due to the fact that Λ is the whole space.
Letp∈D(𝒟) be compactly supported. From Example 2.1 in [5] we know that ϖɛ*p is infinitely differentiable and compactly supported and thus belongs to 𝒴. Equation (28) yields that ϖɛ*p→p as ɛ→0+ in the graph norm of D(𝒟). Because we have shown above that compactly supported functions from D(𝒟) are dense in D(𝒟), we see that (𝒟,D(𝒟)) is the closure of (𝒟,𝒴), and, because T0 is a closed extension of (𝒟,𝒴), we obtain 𝒟⊂T0.
(iii) Lastly we recognize thatD(T0)⊂D(𝒟) so that the operators T0 and 𝒟 coincide, and (G𝒟(t))t≥0=(Z0(t))t≥0. Suppose p∈D(T0). Then for any fixed λ>0 there exists a unique g∈𝒳r such that p=(λI-T0)-1g. For any ψ∈C01(Λ) we have, by (19),
(32)∫Λ𝒟pψdμ=∫Λp(v)(ω·∂ψ)(v)dμv=∫Λ(∫0∞e-λtg(ϕ(v,t,0))dt)(ω·∂ψ)(v)dμv=∫0∞(∫Λe-λtg(ϕ(v,t,0))(ω·∂ψ)(v)dμv)dt=∫0∞(∫Λe-λtg(y)(ω·∂ψ)(ϕ(y,0,t))(v)dμy)dt=∫Λ(∫0∞e-λtddtψ(ϕ(y,0,t))dt)g(y)dμy=∫Λ(λ∫Λ‍(∫0∞‍e-λtψ(ϕ(y,0,t))dt)e-λtψ(ϕ(y,0,t))|0∞g(y)dy+λ∫Λ(∫0∞e-λtψ(ϕ(y,0,t))dt)∫Λ(λ∫Λ‍(∫0∞‍e-λtψ(ϕ(y,0,t))dt)e-λtψ(ϕ(y,0,t))|0∞g(y)dy)g(y)dμy=-∫Λg(y)ψ(y)dμy+λ∫Λ(∫0∞e-λtg(ϕ(v,0,t))dt)ψ(v)dμv=-∫Λ(g-λp)ψdμ.
This implies thatp∈D(𝒟). Hence T0⊂𝒟, and 𝒟p=T0p.Remark 3 (conservativeness of the transport model).
Because the flow process does not modify the total number of individuals in the system, let us show that the model (13) is conservative in the space 𝒳r; that is, the law (4) is satisfied. We have proved that the semigroup generated by the operator 𝒟 is stochastic; then we have
(33)0=∫Λ𝒟pdμ,∀p∈D(𝒟),then0=∫ℝ3∫0∞mr𝒟p(t,x,m)dmdx,∀t≥0,r≥1.
Thus, ∫ℝ3∫0∞m𝒟p(t,x,m)dmdx=0,   for all t≥0 which leads to
(34)ddt𝒩(t)=ddt(∫ℝ3∫0∞mp(t,x,m)dmdx)=∫ℝ3∫0∞m∂tp(t,x,m)dmdx=∫ℝ3∫0∞m𝒟p(t,x,m)dmdx=0
and therefore proving the conservativeness of the transport model in (18).


## 3. Perturbed Transport-Fragmentation Problems

We turn now to the transport problem with the loss part of the fragmentation process. We assume that there are two constants0<θ1 and θ2 such that for every x∈ℝ3,
(35)θ1αm≤a(x,m)≤θ2αm,
with αm∈ℝ+ and independent of x. Then we can consider the loss operator (A,D(A)) defined in (8). The corresponding abstract Cauchy problem reads as
(36)∂tp(t,v)=𝒟p(t,v)-Ap(t,v)=Fp(t,v),p(0,v)=po(v),v∈Λ,
where
(37)F=𝒟-A.
We provide a characterization of the domain D(F).Lemma 4.
ConsiderD(F)=D(𝒟)∩D(A)(=D(𝒟)).Proof.
First of all it is obvious to see thatD(𝒟)∩D(A)=D(𝒟) since D(A)=𝒳r. Because 𝒟 is conservative, integration of (36) over Λ gives (d/dt)∥p∥r=(d/dt)∫ℝ3∫0∞mrp(t,x,m)dmdx=-∫ℝ3∫0∞a(x,m)mrp(x,m)dmdx. Hence (35) leads to
(38)-∫ℝ3∫0∞θ2αmmrp(x,m)dmdx≤-∫ℝ3∫0∞a(x,m)mrp(x,m)dmdx≤-∫ℝ3∫0∞θ1αmmrp(x,m)dmdx
for all p∈(𝒳r)+, and, using Gronwall’s inequality, we obtain
(39)-θ2αm∥p∥r≤ddt∥p∥r≤-θ1αm∥p∥r.
Then
(40)e-θ2αmt∥po∥r≤∥p∥r≤e-θ1αmt∥po∥r.
This inequality for p=GF(t)po yields
(41)e-θ2αmt∥po∥r≤∥GF(t)po∥r≤e-θ1αmt∥po∥r,
where po∈(C0∞(Λ))+⊆D(F)+. If we take 0≤po∈𝒳r, then we can always mollify it by construction of approximations to the identity (mollifiers) ϖɛ(v)=Cɛϖ(v/ɛ) (as in [5, Example 2.1]), where ϖ is a C0∞(Λ) function defined by
(42)ϖ(v)={exp(1|v|2-1)for|v|<1,0for|v|≥1
and Cɛ are constants chosen so that ∫Λϖɛ(v)dx=1.
Using the mollification ofpo by taking the convolution
(43)poɛ:=∫Λpo(v-y)ϖɛ(y)dμy=∫Λpo(y)ϖɛ(v-y)dμy,
we obtain poɛ in 𝒳r (since po∈𝒳r) and po=limɛ→0+poɛin𝒳r. Moreover, poɛ are also nonnegative by (43) since 0≤po, and the family (poɛ)ɛ⊆C0∞(Λ). This shows that any nonnegative po taken in 𝒳r can be approximated by a sequence of nonnegative functions of C0∞(Λ). Inequality (41) is therefore valid for any nonnegative po∈𝒳r. Using the fact that any arbitrary element go of 𝒳r (equipped with the pointwise order almost everywhere) can be written in the form go=go+-go-, where go+,go-∈(𝒳r)+, the positive element approach [18, 19] or [5, Theorem 2.64], allows us to extend the right inequality of (41) to all 𝒳r so as to have
(44)∥GF(t)p∥r≤e-θ1αmt∥p∥r.
Using the semigroup representation of the resolvent [5, Theorem 3.34], we obtain for λ>0(45)∥R(λ,F)p∥r≤∫0∞e-λt∥GF(t)p∥rdt≤∫0∞e-λte-θ1αmt∥p∥rdt≤1λ+θ1αm∥p∥r.
By the right inequality of (35), we obtain that
(46)∥AR(λ,F)p∥r≤θ2αmλ+θ1αm∥p∥r≤θ2θ1∥p∥r.
This relation states that D(A)⊇D(F) (the domain of A is at least that of F). Because F=𝒟-A and A is bounded, we exploit the following relation for resolvent in 𝒳r:(47)λI-F=λI-𝒟+AR(λ,F)(λI-F),I=(λI-𝒟)R(λ,F)+AR(λ,F),R(λ,𝒟)=R(λ,F)+R(λ,𝒟)AR(λ,F),R(λ,F)=R(λ,𝒟)(I-AR(λ,F))
for every m∈ℝ+. This leads to D(𝒟)⊇D(F), and therefore D(F)⊆D(𝒟)∩D(A).
On the other hand, ifp∈D(𝒟)∩D(A) then ∥𝒟p∥r<∞ and ∥Ap∥r<∞. Therefore
(48)∥𝒟p-Ap∥r≤∥𝒟p∥r+∥Ap∥r<∞,
meaning that p∈D(F), and thus D(𝒟)∩D(A)⊆D(F), which ends the proof.By the condition (35), the operator A is the generator of a C0-semigroup of contractions, let us say (GA(t))t≥0. The following theorem holds.Theorem 5.
Assume that (35) is satisfied; then the operator (F,D(F)) is the generator of a substochastic semigroup (GF(t))t≥0 given by
(49)[GF(t)p](v)=[limv→∞[G𝒟(tv)GA(tv)]vp](v)
for p∈𝒳r and t>0, where (G𝒟(t))t≥0 is defined by (21).Proof.
First of all let us prove thatF is the generator of a substochastic semigroup (GF(t))t≥0 in 𝒳r given by
(50)GF(t)p=limv→∞[G𝒟(tv)GA(tv)]vp
for p∈D(F).
We need to show that𝒟 and A satisfy the conditions of Corollary 5.5 in the book by Pazy [20].
(a) We know by Theorem2 and assumption (35) that 𝒟 and A are generators of positive semigroups of contractions; then(51)∥G𝒟(t)∥r≤1=1e0t,∥GA(t)∥r≤1=1e0t∀t≥0.
Thus, 𝒟, A∈𝒢(1,0) and G𝒟(t)≥0,GA(t)≥0 for all t≥0.
(b) By Hille-Yosida Theorem [5, Theorem 3.5], 𝒟 is closed and densely defined in 𝒳r, and because 𝒳r=D(A)⊃D(𝒟), we have D(𝒟)∩D(A)=D(𝒟) is dense in 𝒳r.
(c) By the above condition (a), we can write(52)∥(G𝒟(t)GA(t))v∥r≤∥G𝒟(t)∥rv∥GA(t)∥rv≤1=1e0vt,v=1,2,3,….
(d) By the bounded perturbation theorem [5, Theorem 4.9], 𝒟-A is the generator of a positive semigroup of contractions since 𝒟 generates a positive semigroups of contractions (Theorem 2), and A is bounded (assumption (35)).
We know thatλI-(𝒟-A):D(𝒟)→𝒳r, and by Hille-Yosida Theorem, λI-(𝒟-A) must be invertible for some λ>0 and (λI-(𝒟-A))-1∈ℒ(𝒳r) (the space of bounded linear operators from 𝒳r into 𝒳r). Then the range of λI-(𝒟-A)=𝒳r. Thus λI-(𝒟-A) is densely defined in 𝒳r.
All the conditions of Corollary 5.5 in [20] are satisfied by 𝒟 and A; then F¯=𝒟-A¯=𝒟-A=F is the generator of a semigroup (GF(t))t≥0 defined by
(53)[GF(t)p](v)=[limv→∞[G𝒟(tv)GA(tv)]vp](v),p∈𝒳r,
where we have used the fact that 𝒟-A is closed since it is the generator of a positive semigroup of contractions (Hille-Yosida Theorem).
Let us show that(GF(t))t≥0 is substochastic. By (50) and the above condition (a), we have GF(t)≥0 for all t≥0 since GF(t)p is the limit of a sequence of elements of the positive cone of 𝒳r(54)(𝒳r)+={g∈𝒳r;g≥0}whichisclosed.
Lastly, by (52) and (50), we have
(55)∥GF(t)∥r≤limv→∞∥G𝒟(t)∥rv∥GA(t)∥rv≤1
for all t≥0.Now we take the gain part of the fragmentation process defined by (9) with the coefficients satisfying the conservation law (3) and consider the perturbed transport equation
(56)∂∂tp=𝒟p-Ap+Bp,p|t=0=po.Theorem 6.
If the assumptions of Theorem5 hold, then there is an extension (𝒦,D(𝒦)) of (𝒟-A+B,D(𝒟) that generates the smallest substochastic semigroup on 𝒳r, denoted by (G𝒦(t))t≥0.Proof.
This theorem is a direct continuation of Theorem5 by virtue of the substochastic semigroup theory in Kato’s Theorem in L1 (see [5, Corollary 5.17]). Because D(B):=D(A) (relation (9)), we have D(B)⊃D(𝒟)∩D(A). Thus, to apply Kato’s Perturbation Theorem, we just need to show that, for all p∈D(𝒟-A)+=(D(𝒟))+,
(57)∫Λ(𝒟p-Ap+Bp)dμ≤0.
Since p∈D(𝒟)+ and since ∥Ap∥r<∞,∥Bp∥r<∞, then we can split (57) so as to get its left-hand side equal to
(58)∫Λ𝒟pdμ+∫ℝ3∫0∞mr(∫m∞b(x,s,m)a(x,s)p(x,s)ds-a(x,m)p(t,x,m)+∫m∞b(x,s,m)a(x,s)p(x,s)ds)dmdx.
The first term vanishes by the stochasticity (33) of the operator 𝒟. For the other term, using the relations (11) and (12) yields
(59)∫ℝ3∫0∞mr(∫m∞‍b(x,s,m)a(x,s)p(x,s)ds-a(x,m)p(t,x,m)+∫m∞b(x,s,m)a(x,s)p(x,s)ds∫m∞‍b(x,s,m)a(x,s)p(x,s)ds)dmdx≤∫ℝ3∫0∞mr(-a(x,m)p(t,x,m))dmdx+∫ℝ3∫0∞a(x,s)p(x,s)srdsdx=0,
which proves the theorem.


## 4. Concluding Remarks

In this paper, we used the theory of strongly continuous semigroups of operators [20] to analyze the well posedness of an integrodifferential equation modelling convection-fragmentation processes. This work generalizes the preceding ones with the inclusion of the spatial transportation kernel which was not considered before. We proved that the combined fragmentation-transportation operator is the infinitesimal generator of a strongly continuous stochastic semigroup, thereby addressing the problem of existence of solutions for this model. However the full identification of the generator and characterization of its domain remain an open problem.



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*Source: 320750-2013-06-11.xml*