Topics in the theory of the heat equation 1000-1M23THE

The heat equation is one of the most important objects in the theory of partial differential equations with countless applications to the modeling of real world phenomena. It is also a canonical object that serves as a departure point of a deeper study of more general equations and a testing ground for development of analytical tools. In the standard course of partial differential equations the treatment of the heat flow understandably needs to be abridged to allow for a wealth of other important topics. In this course the heat equation takes center stage. We will cover a range of topics including:
- well-posedness: existence, uniqueness and regularity of solutions under variety of assumptions,
- smoothing properties of the heat flow,
- local and global estimates of solutions,
- long time behaviour of solutions,
- energy, entropy, Fisher information and elements of the optimal transport theory in the context of the heat flow.

Type of course

elective monographs

Mode

Remote learning

Prerequisites (description)

A prospective student should be comfortable with multivariable calculus and analysis in Euclidean spaces. It is advisable but not necessary to have completed a PDE course and a functional analysis course.

Course coordinators

Mikołaj Sierżęga

Course dedicated to a programme

4EU+Courses

Learning outcomes

- Student has a working understanding of the fundamentals of the analysis of the heat equation.
- Student has an understanding of the key characteristics of the heat flow.
- Student has a grasp of the utility the heat equation and its solutions as a tool for more general analytical aims.
- Student has an appreciation of the technical complexities introduced by boundary conditions, singularities and forcing terms.

Assessment criteria

Completion of the course and the final grade will be based on the final project and an overall attendance/performance during the course.

Additional information

Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:

Master's degree, second cycle programme, Mathematics

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 1000-1M23THE in USOSweb