Introductory description

This will be an introduction to variational analysis as opiated with evolution equations. The goal of this course is to introduce some fundamental concepts, methods and theory associated with the mathematical theory of time dependent PDEs and related models. The essence will be the abstract theory of variational formulations of parabolic and second order evolution equations and the theory of gradient flows. Motivation comes from physical, life and social sciences.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1.Abstract formulation of linear equations : Hille-Yosida Theorem
2. Gradient flows
3. PDE examples: Heat and wave equation
4. Applications

Learning outcomes

By the end of the module, students should be able to:

• - Have a grasp of the variational theory of evolution equations and gradient flows
• - Understand and apply the Hille-Yosida theorem
• - Formulate PDEs in a variational framework
• - Recognise gradient flow
• - Apply the Galerkin and Rothe method for well posedness
• - Carry out variational analysis in a variety of settings
• -Acquire some knowledge of applications

Indicative reading list

There will be typed lecture notes. There will be material related to chapters in the following:-
H. Brezis Functional analysis, Sobolev spaces and Partial Differential Equations Springer Universitext (2011)
L. C. Evans Partial Differential Equations AMS Grad Studies in Maths Vol 19
Michel Chipot Elements of nonlinear analysis Birkhauser Advanced Texts (2000)
H. Attouch, G.Butazzo, G. Michaille Variational analysis in Sobolev and BV spaces: Applications to PDEs and optimization SIAM (2014)
S. Bartels Numerical methods for nonlinear PDEs Springer (2015)

Subject specific skills

The course will expose students to a wide range of ideas in an advanced subject at the forefront of current mathematical research. The students will learn to familiarise themselves with abstract concepts, and to relate them to physical intuition.

Transferable skills

They will learn to follow complex reasoning, to construct logical arguments, to think independently, and to develop research skills.