Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 320750, 8 pages

http://dx.doi.org/10.1155/2013/320750

Research Article

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

S. C. Oukouomi Noutchie and E. F. Doungmo Goufo

Department of Mathematical Sciences, North-West University, Mafikeng 2735, South Africa

Received 27 March 2013; Accepted 23 May 2013

Academic Editor: Guo-Cheng Wu

Copyright © 2013 S. C. Oukouomi Noutchie and E. F. Doungmo Goufo. 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.

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.

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]:

where, in terms of the mass size and the position , the state of the system is characterized at any moment by the particle-mass-position distribution ( is also called the density or concentration of particles), with . The three-dimensional vector represents the velocity of the transport and is supposed to be a known quantity depending on and ; is the average fragmentation rate; that is, it describes the ability of aggregates of size and position to break into smaller particles. Once an aggregate of mass and position breaks, the expected number of daughter particles of size is the nonnegative measurable function defined on . The space variable is supposed to vary in the whole of . The function represents the density of groups of size at position at the beginning ().

2.1. Fragmentation Equation

Let us introduce necessary assumptions that will be useful in our analysis. Since a group of size cannot split to form a group of size , the function has its support in the set

After the fragmentation of a mass particle, the sum of masses of all daughter particles should again be ; hence it follows that, for any ,

Because the space variable varies in the whole of (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:

where is the total number of individuals in the space (or total mass of the ensemble). Since is the density of groups of size at the position and time and that mass is expected to be a conserved quantity, the most appropriate Banach space to work in is the space

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):

which coincides with for and is endowed with the norm . We assume that , and, for each , the function is from the space with . When any subspace , we will denote by the subset of defined as . Note that any will possess moments

of all orders . In , we define from the expressions on the right-hand side of (1) the operators and by

Lemma 1. is a well-defined operator.

Proof. To prove that is well defined on as stated in (9), we use the condition (3) to show that

Hence

for . Note that the equality holds for . For every , changing the order of integration by the Fubini theorem, we have

where we have used inequality (11). The result follows from the fact that any arbitrary element of can be written in the form , where . Then , for all , so that we can take , and is well defined.

2.2. Cauchy Problem for the Transport Operator in

is endowed with the Lebesgue measure . Our primary objective in this section is to analyze the solvability of the transport problem

in the space .

Furthermore, to simplify the notation we put . We consider the function and the expression appearing on the right-hand side of (13). Then

We assume that is divergence-free and globally Lipschitz continuous. Then , and (14) becomes

For and , the initial value problem

has one and only one solution taking values in . Thus we can consider the function defined by the condition that, for ,

is the only solution of the Cauchy problem (16). The integral curves given by the -parameter family (with , the only solution of (16)) are called the characteristics of . The function possesses many desirable properties [13–15] that will be relevant for studying the transport operator in . Some of them are listed in [5, Proposition 10.1]. Now we can properly study the transport operator . Using the above proposition in our application, we can take

Note that is understood in the sense of distribution. Precisely speaking, if we take as the set of the test functions, each if and only if , and there exists such that

for all , where

with , the th component of the velocity . The middle term in (19) exists as is globally Lipschitz continuous, and the last equality follows as is divergence-free. If this is the case, we define .

Now we can show that the operator is the generator of a stochastic semigroup on .

Theorem 2. If the function is globally Lipschitz continuous and divergence-free, then the operator defined by (18) is the generator of a strongly continuous stochastic semigroup , given by

for any and .

Proof. Let 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 is a strongly continuous semigroup of bounded linear operators. We need some properties of as listed in [5] and given as follows. The function has the following properties:

for all ; for all , and ; for all ; for all ; function is continuous; the transformation defined by , and is a topological homeomorphism which is bimeasurable, and its inverse is represented by , and ; for all the transformation of onto itself defined by is measure preserving.

Then by the properties () and (), we see that, for any , the composition , in (21), is a measurable function satisfying the equality

Hence the family is of bounded linear operators from . Then we can easily verify the following relations:

; , for all ;, for each .

In fact, () and () follow immediately from the properties () and (). To prove (), we can follow the argument of Example 3.10 in [5]. Thus, it is enough to show () for . For such , we have for all . Furthermore, if for all , then for all , and, because the support of is bounded, the Lebesgue dominated convergence theorem shows that () is satisfied. Thus is a -semigroup.

(ii) Secondly, we prove that the generator of is an extension of .

Let be the set of real-valued functions which are defined on , are Lipschitz continuous, and compactly supported. Obviously because if , then the first-order partial derivatives of are measurable, bounded, and compactly supported and thus, multiplied by Lipschitz continuous functions of , belong to . For any fixed , we denote by the real-valued function defined on by

From the previous considerations and properties ()–() there exists a measurable subset of , with , such that at each point the function has measurable first-order partial derivatives. In particular,

and, therefore, if we let , then

for any . From this and from part of the proof it follows that, for every ,

as . This proves that and that , for all . Next we prove that is a core of , that is, that is the closure of . Let , be a mollifier (see Example 2.1 in [5]), and, for , let be the mollification of . We use the Friedrichs lemma, [16, pp. 313–315], or [17, Lemma ], which states that there is , independent of , such that for any function , we have

Estimates of Equation in [5] and the above relation (27) imply

which shows that the mollification is a continuous operator in (equipped with the graph norm) uniformly bounded with respect to . Next we observe that the subset of consisting of compactly supported functions is dense in with the graph norm. Indeed, let . Because both , the absolute continuity of the Lebesgue integral implies that for any given there exists a compact subset of such that

For this we choose satisfying for all , and for all . Now it is easy to see that and has a compact support. Moreover,

where can be made independent of due to the fact that is the whole space.

Let be compactly supported. From Example 2.1 in [5] we know that is infinitely differentiable and compactly supported and thus belongs to . Equation (28) yields that as in the graph norm of . Because we have shown above that compactly supported functions from are dense in , we see that is the closure of , and, because is a closed extension of , we obtain .

(iii) Lastly we recognize that so that the operators and coincide, and . Suppose . Then for any fixed there exists a unique such that . For any we have, by (19),

This implies that . Hence , and .

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 ; that is, the law (4) is satisfied. We have proved that the semigroup generated by the operator is stochastic; then we have

Thus, ,   for all which leads to

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 constants and such that for every ,

with and independent of . Then we can consider the loss operator defined in (8). The corresponding abstract Cauchy problem reads as

where

We provide a characterization of the domain .

Lemma 4. Consider .

Proof. First of all it is obvious to see that since . Because is conservative, integration of (36) over gives . Hence (35) leads to

for all , and, using Gronwall’s inequality, we obtain

Then

This inequality for yields

where . If we take , then we can always mollify it by construction of approximations to the identity (mollifiers) (as in [5, Example 2.1]), where is a function defined by

and are constants chosen so that .

Using the mollification of by taking the convolution

we obtain in (since ) and . Moreover, are also nonnegative by (43) since , and the family . This shows that any nonnegative taken in can be approximated by a sequence of nonnegative functions of . Inequality (41) is therefore valid for any nonnegative . Using the fact that any arbitrary element of (equipped with the pointwise order almost everywhere) can be written in the form , where , the positive element approach [18, 19] or [5, Theorem 2.64], allows us to extend the right inequality of (41) to all so as to have

Using the semigroup representation of the resolvent [5, Theorem 3.34], we obtain for

By the right inequality of (35), we obtain that

This relation states that (the domain of is at least that of ). Because and is bounded, we exploit the following relation for resolvent in

for every . This leads to , and therefore .

On the other hand, if then and . Therefore

meaning that , and thus , which ends the proof.

By the condition (35), the operator is the generator of a -semigroup of contractions, let us say . The following theorem holds.

Theorem 5. Assume that (35) is satisfied; then the operator is the generator of a substochastic semigroup given by

for and , where is defined by (21).

Proof. First of all let us prove that is the generator of a substochastic semigroup in given by

for .

We need to show that and satisfy the conditions of Corollary 5.5 in the book by Pazy [20].

(a) We know by Theorem 2 and assumption (35) that and are generators of positive semigroups of contractions; then

Thus, , and for all .

(b) By Hille-Yosida Theorem [5, Theorem 3.5], is closed and densely defined in , and because , we have is dense in .

(c) By the above condition (a), we can write

(d) By the bounded perturbation theorem [5, Theorem 4.9], is the generator of a positive semigroup of contractions since generates a positive semigroups of contractions (Theorem 2), and is bounded (assumption (35)).

We know that , and by Hille-Yosida Theorem, must be invertible for some and (the space of bounded linear operators from into ). Then the range of . Thus is densely defined in .

All the conditions of Corollary 5.5 in [20] are satisfied by and ; then is the generator of a semigroup defined by

where we have used the fact that is closed since it is the generator of a positive semigroup of contractions (Hille-Yosida Theorem).

Let us show that is substochastic. By (50) and the above condition (a), we have for all since is the limit of a sequence of elements of the positive cone of

Lastly, by (52) and (50), we have

for all .

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

Theorem 6. If the assumptions of Theorem 5 hold, then there is an extension of that generates the smallest substochastic semigroup on , denoted by .

Proof. This theorem is a direct continuation of Theorem 5 by virtue of the substochastic semigroup theory in Kato’s Theorem in (see [5, Corollary 5.17]). Because (relation (9)), we have . Thus, to apply Kato’s Perturbation Theorem, we just need to show that, for all ,

Since and since , then we can split (57) so as to get its left-hand side equal to

The first term vanishes by the stochasticity (33) of the operator . For the other term, using the relations (11) and (12) yields

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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