International Max Planck Research School for Analysis, Design and Optimization in Chemical and Biochemical Process Engineering Magdeburg " The Singular Coagulation and Coagulation-Fragmentation Equations"

Certain problems in the physical sciences are governed by the coagulation and the coagulation-fragmentation equations. These equations are a type of integro-differential equations which are also known as aggregation and aggregation-breakage equations respectively. The coagulation (aggregation) term describes the kinetics of particle growth where particles can coagulate (aggregate) to form larger particles via binary interaction. On the other side, the fragmentation (breakage) term describes how particles break into two or more fragments. The term aggregation covers two processes, the coagulation and agglomeration process. The coagulation process is when particles aggregate forming a new particle where it is not possible to define them in the new particle. The agglomeration process is when particles aggregate and it is possible to define them in the new particle. The coagulation and agglomeration processes are often found in liquid and solid substance respectively. Mathematically the two processes are described by the same equation, therefore we will refer to it as coagulation.Breakage and fragmentation are also synonyms. In many applications, the size of a particle is considered as the only relevant particle property. If we describe the size of a particle by its mass, we have that during the coagulation process the total number of particles decreases while by the fragmentation process the total number of particles increases. In the coagulation process as well as in the fragmentation process the total mass remains constant. Examples of these processes can be found e.g. in astrophysics , in chemical and process engineering, polymer science, and aerosol science.

The aim of this work was to present some results related to the existences and uniqueness of solutions to the coagulation and the coagulation equation with multifragmentation.

We presented a proof of an existence theorem of solutions to the Smoluchowski coagulation equation for a very general class of kernels. This class of kernels includes singular kernels. The important Smoluchowski coagulation kernel for Brownian motion, the equi-partition of kinetic energy (EKE) kernel, and the granulation kernel are covered by our analysis. Our result is obtained in a suitable weighted Banach space of L^1 functions. We define a sequence of truncated problems from our original problem in order to eliminate the singularities of the kernels. Using the contraction mapping principle, we proved the existence and uniqueness of solutions to them. Using weak compactness theory, we prove that this sequence of solutions converges to a certain function. Then it was shown that the limiting function solves the original problem. The uniqueness result was obtained by taking the difference of two solutions and showing that this difference is equal to zero by appliying Gronwall’s inequality.

Using the same technique we proved the existence and uniqueness of solutions to the singular coagulation equation with multifragmentation in a suitable weighted Banach space of L^1 functions extending the previous result. The Smoluchowski coagulation kernel for Brownian motion, the equi-partition of kinetic energy (EKE) kernel, and the granulation kernel are examples of singular coagulation kernels which are covered in our analysis. It is important to point out that there is no previous existence result mentioned kernels of solution to the coagulation-fragmentation equation with singular kernel.