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

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.


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][2][3][4] and references therein).Various types of pure fragmentation equations have been comprehensively analyzed in numerous works (see, e.g., [5][6][7][8][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.

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  : R + × R 3 × R + → R + .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 R 3 × R × R. The space variable  is supposed to vary in the whole of R 3 .The function   (, ) represents the density of groups of size  at position  at the beginning ( = 0).

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  > 0,  ∈ R 3 Because the space variable  varies in the whole of R 3 (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 N() = ∫ R 3 ∫ ∞ 0 (, , ) 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 X 1 for  = 1 and is endowed with the norm ‖ ⋅ ‖  .We assume that o  ∈ X  , and, for each  ≥ 0, the function (, ) → (, ) = (, , ) is from the space X  with  ≥ 1.When any subspace  ⊆ X  , we will denote by  + the subset of  defined as  + = { ∈ ; (, ) ≥ 0,  ∈ R + ,  ∈ R 3 }.Note that any  ∈ (X  ) + will possess moments of all orders  ∈ [0, ].In X  , 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  ≥ 1,  > 0. Note that the equality holds for  = 1.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.

Cauchy Problem for the Transport Operator in
Λ is endowed with the Lebesgue measure  =  , = .Our primary objective in this section is to analyze the solvability of the transport problem in the space X  .Furthermore, to simplify the notation we put k = (, ) ∈ Λ.We consider the function  : Λ → R 3 and D the expression appearing on the right-hand side of (13).Then We assume that  is divergence-free and globally Lipschitz continuous.Then div (k) := ∇ ⋅ (k) = 0, and ( 14) becomes For k ∈ Λ and  ∈ R, the initial value problem has one and only one solution r() taking values in Λ.Thus we can consider the function  : Λ × R 2 → Λ defined by the condition that, for (k, ) ∈ Λ × R, is the only solution of the Cauchy problem (16).The integral curves given by the -parameter family (r)  (with r() = (k, , ),  ∈ R, the only solution of ( 16)) are called the characteristics of D. The function  possesses many desirable properties [13][14][15] that will be relevant for studying the transport operator in X  .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 D = D, with D represented by (15) , Note that D is understood in the sense of distribution.
Precisely speaking, if we take  1 0 (Λ) as the set of the test functions, each  ∈ (D) if and only if  ∈  1 (Λ), and there exists  ∈ X  such that for all  ∈  1 0 (Λ), where with   =   (k), the th component of the velocity (k).
The middle term in (19) exists as  is globally Lipschitz continuous, and the last equality follows as  is divergencefree.If this is the case, we define D = .Now we can show that the operator D is the generator of a stochastic semigroup on X  .Theorem 2. If the function  is globally Lipschitz continuous and divergence-free, then the operator ((D), D) defined by (18) is the generator of a strongly continuous stochastic semigroup ( D ()) ≥0 , given by for any  ∈ X  and  > 0.
Proof.Let ( 0 ()) ≥0 be the family defined by the right-hand side of the relation (21).The proof of the theorem will follow three steps.
(ii) Secondly, we prove that the generator  0 of ( 0 ()) ≥0 is an extension of D.
Let Y be the set of real-valued functions which are defined on Λ, are Lipschitz continuous, and compactly supported.Obviously Y ⊂ (D) because if  ∈ Y, then the first-order partial derivatives of  are measurable, bounded, and compactly supported and thus, multiplied by Lipschitz continuous functions of , belong to  1 (Λ, ).For any fixed  ∈ Y, we denote by  the real-valued function defined on Λ ×  + by  (k, ) = ( 0 () ) (k) . ( From the previous considerations and properties ( 3 )-( 5 ) there exists a measurable subset  of Λ ×  + , with (Λ ×  + \ ) = 0, such that at each point (k, ) ∈  the function  has measurable first-order partial derivatives.In particular, where  = sup |D| can be made independent of Ω  due to the fact that Λ is the whole space.
Let  ∈ (D) be compactly supported.From Example 2.1 in [5] we know that   *  is infinitely differentiable and compactly supported and thus belongs to Y. Equation (28) yields that   *  →  as  → 0 + in the graph norm of (D).Because we have shown above that compactly supported functions from (D) are dense in (D), we see that (D, (D)) is the closure of (D, Y), and, because  0 is a closed extension of (D, Y), we obtain D ⊂  0 .
Suppose  ∈ ( 0 ).Then for any fixed  > 0 there exists a unique  ∈ X  such that  = (− 0 ) −1 .For any  ∈  1 0 (Λ) we have, by (19), This implies that  ∈ (D).Hence  0 ⊂ D, and D =  0 .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 X  ; that is, the law (4) is satisfied.We have proved that the semigroup generated by the operator D is stochastic; then we have and therefore proving the conservativeness of the transport model in (18).

Perturbed Transport-Fragmentation Problems
We turn now to the transport problem with the part of the fragmentation process.We assume that there are two constants 0 <  1 and  2 such that for every  ∈ R 3 , with   ∈ R + 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 ().
By the condition (35), the operator  is the generator of a  0 -semigroup of contractions, let us say (  ()) ≥0 .The following theorem holds.
Proof.First of all let us prove that  is the generator of a substochastic semigroup (  ()) ≥0 in X  given by for  ∈ ().
We need to show that D and  satisfy the conditions of Corollary 5.5 in the book by Pazy [20].We know that  − (D − ) : (D) → X  , and by Hille-Yosida Theorem,  − (D − ) must be invertible for some  > 0 and ( − (D − )) −1 ∈ L(X  ) (the space of bounded linear operators from X  into X  ).Then the range of  − (D − ) = X  .Thus  − (D − ) is densely defined in X  .
[5]m this and from part (i) of the proof it follows that, for every ℎ > 0, + .This proves that Y ⊂ ( 0 ) and that  0  = D, for all  ∈ Y. Next we prove that Y is a core of D, that is, that (D, (D)) is the closure of (D, Y).Let   ,  > 0, 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 1.2.5], which states that there is  > 0, independent of , such that for any   function , 1 ≤  < ∞, we have     D (  * ) −   * D ∈ Λ, and (k) = 1 for all k ∈ Ω  .Now it is easy to see that  ∈ (D) and has a compact support.Moreover, and, therefore, if we let   := ess sup (k)∈Λ |D|, then       (k, )     ≤   (25)for any (k, ) ∈ .whichshows that the mollification  →   *  is a continuous operator in (D) (equipped with the graph norm) uniformly bounded with respect to .Next we observe that the subset of (D) consisting of compactly supported functions is dense in (D) with the graph norm.Indeed, let  ∈ (D).Because both , D ∈ X  , the absolute continuity of the Lebesgue integral implies that for any given  > 0 there exists a compact subset Ω  of Λ such that ∫ − 1            − 2                      , then we can always mollify it by construction of approximations to the identity (mollifiers)   (k) =   (k/) (as in [5, Example 2.1]), where  is a  ∞ 0 (Λ) function defined by  (equipped with the pointwise order almost everywhere) can be written in the form  ) + , the positive element approach [18, 19] or [5, Theorem 2.64], allows us to extend the right inequality of (41) to all X  so as to have       ()      ≤  − 1            .R + .This leads to (D) ⊇ (), and therefore () ⊆ (D) ∩ ().On the other hand, if  ∈ (D) ∩ () then ‖D‖  < ∞ and ‖‖  < ∞.