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.
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 [
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 [
We consider the following Cauchy problem [
Let us introduce necessary assumptions that will be useful in our analysis. Since a group of size
To prove that
Furthermore, to simplify the notation we put
Note that
Now we can show that the operator
If the function
Let
(i) First we show that function the transformation for all
Then by the properties (
In fact, (
(ii) Secondly, we prove that the generator
Let
where
Let
(iii) Lastly we recognize that
This implies that
Because the flow process does not modify the total number of individuals in the system, let us show that the model (
We turn now to the transport problem with the loss part of the fragmentation process. We assume that there are two constants
Consider
First of all it is obvious to see that
Using the mollification of
On the other hand, if
By the condition (
Assume that (
First of all let us prove that
We need to show that
(a) We know by Theorem
(b) By Hille-Yosida Theorem [
(c) By the above condition (a), we can write
(d) By the bounded perturbation theorem [
We know that
All the conditions of Corollary 5.5 in [
Let us show that
Now we take the gain part of the fragmentation process defined by (
If the assumptions of Theorem
This theorem is a direct continuation of Theorem
In this paper, we used the theory of strongly continuous semigroups of operators [