Coagulation dynamics

This course is aimed at Master and PhD students with background in Mathematics, Physics or Aerosol Modelling.

The coagulation (coalescence or aggregation) of small particles forming bigger ones is a process that can be observed in physical systems, e.g., aerosol and raindrop formation, smoke, sprays and galaxies, as well as in biological systems, e.g., hematology, bacteria aggregation and animal grouping. The evolution of the particle size distribution can be described by the classical Smoluschowki's coagulation equation, which is an integro-differential equation belonging to the class of Kinetic Equations and also to the class of General Dynamics Equations. The solutions exhibit rich behavior depending on the rate of coagulation considered, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time).

We will study existence and uniqueness of solutions, explicitly solvable kernels and properties of the solutions such as mass conservation, gelation, self-similar profiles and long-time behavior.

The recommended prerequisite is differential calculus, but the course will be mainly self-contained.

Enrol

Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 13.1.2020
16:15 - 18:00
Tue 14.1.2020
14:15 - 16:00
Thu 16.1.2020
16:15 - 18:00
Mon 27.1.2020
16:15 - 18:00
Tue 28.1.2020
14:15 - 16:00
Thu 30.1.2020
16:15 - 18:00
Mon 3.2.2020
16:15 - 18:00
Tue 4.2.2020
14:15 - 16:00
Mon 10.2.2020
16:15 - 18:00
Tue 11.2.2020
14:15 - 16:00
Mon 17.2.2020
16:15 - 18:00
Tue 18.2.2020
14:15 - 16:00
Mon 24.2.2020
16:15 - 18:00
Tue 25.2.2020
14:15 - 16:00

Other teaching

15.01.2020 Wed 12.15-14.00
29.01. - 26.02.2020 Wed 12.15-14.00
06.02.2020 Thu 16.15-18.00
Marina Amado Ferreira
Teaching language: English

Material

Tasks

Exercise sheet 1

Exercise sheet 3

Exercise sheet 3 - corrected

I added some steps in the last exercise (ex 5).

Exercise 4

Exercise 5

Exercise sheet 6

Description

Optional course.

Master’s Programme in Theoretical and Computational Methods is responsible for the course.

It belongs to the TCM300 Advanced Studies in Theoretical and Computational Methods module.

The course is available to students from other degree programmes.

Differential/integral calculus and partial differential equations.

Kinetic theory, aerosol modelling.

Learning analytic methods for integro-differential equations and how to apply them to coagulation equations.
Understanding the relation between different classes of coagulation kernels and certain phenomena such as mass conservation, gelation and self-similar behaviour.
Realizing about the state of art of the topic, including open problems and applications to aerosol dynamics.

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Existence and uniqueness of solutions, explicit solutions, mass conservation and gelation, long-time behavior (self-similar solutions and stability).

Project work

Course lecture notes.

Recommended:

Dubovskii P. B., "Mathematical Theory of Coagulation."

Laurençot, P., "Weak compactness techniques and coagulation equations." Evolutionary equations with applications in natural sciences. Springer, Cham, 2015. 199-253.

Friendlander, S. K. (2000). "Smoke, dust and haze: Fundamentals of aerosol dynamics."

Lectures and exercise classes with bonus points.

Course will be graded with grades 1-5.

This course is aimed at Master and PhD students with background in Mathematics or Physics who are willing to learn analytic tools for integro-differential equations and to apply them to the study of coagulation dynamics.

The coagulation (coalescence or aggregation) of small particles forming bigger ones is a process that can be observed in physical systems, e.g., aerosol and raindrop formation, smoke, sprays and galaxies, as well as in biological systems, e.g., hematology, bacteria aggregation and animal grouping. The evolution of the particle size distribution can be described by the classical Smoluschowki's coagulation equation, which is an integro-differential equation belonging to the class of Kinetic Equations and also to the class of General Dynamics Equations. The solutions exhibit rich behavior depending on the rate of coagulation considered, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time).

We will study existence and uniqueness of solutions, explicitly solvable kernels and properties of the solutions such as mass conservation, gelation, self-similar profiles and long-time behavior.