# MA4L9 Variational Analysis and Evolution Equations

**Lecturer: **Charles Elliott

**Term(s): **2

**Status for Mathematics students: **

**Commitment: **10 x 3 hour lectures + 9 x 1 hour support classes

**Assessment: **3 hour written examination 100%

**Prerequisites: **MA3G7 Functional Analysis, MA3G1 Theory of PDEs Strongly recommended to have taken MA359 Measure Theory. Ideally one would take MA4A2 Advanced PDEs and have a background in Sobolev spaces.

**Leads To:
** Graduate studies in PDEs. Links to advanced courses in analysis, PDEs, stochastic analysis, numerical analysis, applications.

**Content:**

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 of the main material behind the most common evolutionary PDEs. In particular, the focus will be on functional analytical approaches to find well posed formulations and properties of their solutions.

This course is particularly suitable for students who have liked analysis and differential equation courses in earlier years and to students interested in applications of mathematics. Many students intending graduate studies will find it useful. 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. Topics include

1.Abstract formulation of linear equations : Hille-Yosida Theorem, Lions-Lax-Milgram Theorem

2. PDE examples: Heat and wave equation

3. Gradient flows

4. Applications

**Aims:**

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.

**Objectives:**

**By the end of the module the student 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

**Books:** 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)*