# Variational analysis and evolution equations

TCC MODULE
PHD STUDENTS (and others) FROM
BATH-BRISTOL-ICL-OXFORD-WARWICK:
FOR OFFICIAL REGISTRATION PLEASE FOLLOW THE INSTRUCTIONS ON THE TCC WEB PAGE:-

ALL LECTURES WILL BE HELD VIA MS TEAMS FOR REGISTERED STUDENTS

All lectures in this module start on the week beginning 12 October 2020 and run for 8 consecutive weeks on
Monday 11:00 til 13:00
LECTURE NOTES
The "live" handwritten lecture notes will be posted on the MSTeams site. after the lecture.
Typed lecture notes will also appear there in a draft form which will be updated regularly.
OUTLINE

Because of the ubiquitous nature of PDE based mathematical models in biology, advanced materials, data analysis, finance, physics and engineering, much of mathematical analysis is devoted to their study. Often the models are time-dependent; the state evolves in time. Although the complexity of the models means that finding formulae for
solutions is impossible in most practical situation one can develop a functional analysis based framework for establishing well-posedness in a variety of situations.

This course covers some fundamental concepts, methods and theory associated with the variational theory of time-dependent PDEs and related models. In particular, the focus will be on functional analysis techniques to find abstact well-posed variational formulations of parabolic and second-order evolution equations and the theory of gradient flows. Motivation comes from physical, life and social sciences.

This course is suitable for students with interests in analysis and differential equations and to students interested in applications of mathematics. It will be useful for many preparing for a PhD, and essential for others. There are not too many prerequisites, although you will need some functional analysis, some knowledge of measure theory and an acquaintance with partial differential equations.

Content:

Topics include

1. Background functional analysis

2. Necessary Sobolev spaces and elliptic equation theory

3. Abstract formulation of linear equations :

Lions-Lax-Milgram Theorem

Hille-Yosida Theorem,

4. PDE examples: Heat and wave equation