Coagulation-Fragmentation Equations and Semigroups: A meeting in Honour of Wilson Lamb, 23rd May, 2018

The Applied Analysis Group, supported by the Department of Mathematics and Statistics and the Edinburgh Mathematical Society, are pleased to announce a one-day conference on Coagulation-Fragmentation Equations and Semigroups in honour of Dr Wilson Lamb who recently retired from the University of Strathclyde.

"Wilson Lamb's research career started with his PhD thesis, supervised by Adam McBride at the University of Strathclyde, which was entitled Fractional Powers of Operators on Fréchet Spaces with Applications. It extended known results based on spectral theory in Banach Spaces. This philosophy has underpinned much of his subsequent work which has been conducted within the setting of spaces of test functions and the corresponding spaces of generalised functions. Since fractional powers of an operator form a semigroup under composition, it was a natural progression to study semigroups of operators and evolution equations in Fréchet spaces. A particular focus for over 20 years has been provided by models for coagulation and fragmentation. The use of generalised functions has enabled rigorous mathematical analysis of such models, leading to results on existence and uniqueness of solutions of different types. Perturbation methods for semigroups of operators in Banach spaces have also played a prominent role. Wilson's work has attracted the interest of others working in the field, notably Jacek Banasiak and Philippe Laurençot with both of whom he is currently writing a monograph.

Wilson has 49 refereed research publications and one refereed book chapter. He has supervised 12 PhD students and one MPhil student.  He has given presentations in many countries, notably in South Africa where he has been a member of the Associate Faculty of the African Institute for Mathematical Sciences ( AIMS ) since 2012 and an Honorary Professor at the University of KwaZulu-Natal in Durban since 2013.

Wilson was a wonderful colleague at Strathclyde for 36 years. He was a superb teacher, hugely popular with students for his approachability and willingness to help. He performed many administrative roles, all of which were carried out in exemplary fashion. To many Wilson Lamb is an unsung hero. The present conference is fitting recognition of an outstanding career."   Adam McBride, Prof. Emeritus, University of Strathclyde.


9:45 - 10:00  
Opening and welcome: Tony Mulholland (Head of Department)

10:00 - 10:15
Wilson's Scientific Work: Adam McBride OBE

10:15 - 11:00
Discrete Coagulation-Fragmentation Systems: Lyndsay Kerr

11:00 - 11:30
Tea/Coffee Break

11:30 - 12:15 
Discrete Coagulation-Fragmentation Processes with Growth and Decay: JacekBanasiak

12:15 - 13:00
Numerical Nightmares in Discrete Coagulation-Fragmentation Dynamics: Dugald Duncan

13:00 - 14:30

14:30 - 15:15
Self-similar Solutions to Coagulation-Fragmentation Equations: Philippe Laurençot

15:15 - 16:00
Non-self-similar Behaviour in Smiluchowski's Coagulation Equation: Barbara Niethammer

16:00 - 16:30
Tea/Coffee Break

16:30 - 17:15
Coagulation-Fragmentation Group Dynamics via Bernstein Function Theory: Robert Pego

17:15 - 17:30
Closing Address: Wilson Lamb

Costs and booking

The costs are: £25 for all external attendees which covers the cost of lunch and tea/coffee breaks, free to PhD students as well as past and present University of Strathclyde staff.

There are 50 places available and booking is through the university online shop. 

Please contact us if you need more information.

Dugald B. Duncan (Heriot Watt University): Numerical Nightmares in Discrete Coagulation-Fragmentation Dynamics

We discuss various challenges arising in the numerical simulation of discrete coagulation-fragmentation equations.  These include coping with metastability and very large system sizes in the Becker-Döring equations, as well as trying to capture gelation in more general coagulation-fragmentation systems. 


Most of this work was either inspired by or done jointly with Jack Carr, and the work on gelation relies on his analysis of the problem using centre manifold techniques.

Robert L. Pego (Carnegie-Mellon University): Coagulation-Fragmentation Group Dynamics via Bernstein Function Theory

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish.  The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with n nodes. The structure of equilibrium profiles and other related sequences is explained through an elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on a half line. 


This is joint work with Jian-Guo Liu and Pierre Degond.

Jacek Banasiak (University of Pretoria): Discrete Coagulation-Fragmentation Processes with Growth and Decay

Coagulation and fragmentation models that describe objects forming larger clusters or, conversely, splitting into smaller fragments, have received a lot of attention over several decades due to their importance in chemical engineering and other fields of science and technology. One of the most efficient approaches to modelling dynamics of such processes is through the kinetic (rate) equation which describes the evolution of the distribution of interacting clusters with respect to their size/mass. More recently it has been observed that also living organisms form clusters or split into subgroups depending on circumstances. It concerns large animals forming herds, fish forming schools, but also phytoplankton, or cell division models. However, living matter has its own vital dynamics; that is, in addition to forming or breaking clusters, individuals within them are born or die and so the latter processes must be adequately represented in the models. In discrete models they are modelled by adding the classical birth-and-death terms to the Smoluchowski equation.


In the talk we shall describe how the substochastic semigroup theory can be used to prove analyticity and compactness of a large class of fragmentation semigroups. This result is applied to discrete fragmentation processes with growth to analyze their long time behaviour and to prove the existence of classical solutions to equations describing processes that combining the vital processes with fragmentation and coagulation.

Philippe Laurençot (University of Toulouse): Self-similar Solutions to Coagulation-Fragmentation Equations

When the coagulation kernel and the overall fragmentation rate are homogeneous, there is a specific value of both homogeneity orders which separates two different behaviours: all solutions are expected to be mass-conserving in one case while gelation is expected to take place in the other, provided the mass of the initial condition is large enough.


The focus of this talk is the critical case for which we establish the existence of mass-conserving self-similar solutions, possibly for small masses. This is partly a joint work with Henry van Roessel (Edmonton).

Barbara Niethammer (University of Bonn): Non-self-similar Behaviour in Smoluchowski's Coagulation Equation

Smoluchowski's coagulation equation is a nonlocal integral equation that is used to describe such diverse aggregation phenomena  as fog formation, polymerization, soot agglomeration or even the formation of stars.

The rate at which aggregation takes place is determined by a rate kernel that subsumes the details of the aggregation process under consideration.

It has been conjectured in the applied literature that the long-time behaviour of solutions is described by self-similar solutions. This has been proved to be true for the small class of solvable kernels. In the general case this issue is mainly open, but we will present an example that shows that for kernels that concentrate near the diagonal the large-time behaviour is not self-similar.  (Based on joint work with M. Bonacini, M. Herrmann and J.J.L. Velazquez) 

Lyndsay Kerr (University of Strathclyde): Discrete Coagulation-Fragmentation Systems

In many situations in nature and industrial processes, clusters of particles can combine into larger clusters or fragment into smaller clusters. The evolution of these clusters can be described by di erential equations known as coagulation-fragmentation equations. In the discrete size case it is assumed that the mass of each cluster is a natural number and a cluster of mass n consists of n identical units.

The main part of the talk will concentrate on the case of pure discrete fragmentation. The fragmentation system will be written as an abstract Cauchy problem, in the most physically relevant Banach space, and the theory of substochastic C0 -semigroups will be used to obtain results relating to the existence of a unique, positive, mass conserving solution. Moreover, the existence of solutions in other weighted ℓ1 spaces shall also be examined using perturbation results which give rise to the existence of analytic semigroups. The full coagulation-fragmentation system, where the coagulation coecients may be time-dependent, will also be briefly examined.

Joint work with Wilson Lamb and Matthias Langer.