# MA4L9 Variational Analysis and Evolution Equations

**Lecturer: **Charles Elliott

**Term: **Term 1

**Status for Mathematics students: **List C

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

**Assessment: ** 100% Examination

**Formal registration prerequisites: **None

**Assumed knowledge:**

- MA3G7 Functional Analysis I Banach and Hilbert spaces, Dual spaces, Linear operators, Riesz Representation Theorem
- MA359 Measure Theory Lebesgue integration, properties and convergence theorems
- MA259 Multivariable Calculus Differentiable functions of several variables, Divergence theorem

**Useful background: **

**Synergies: **

The module fits well with other modules in Analysis and Applications involving partial differential equations, particularly MA4A2 Advanced PDEs. Essential or useful for research in much of analysis, dynamical systems, probability and applied mathematics etc.

**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:

- Abstract formulation of linear equations, Bochner spaces
- Hille-Yosida Theorem, Lions-Lax-Milgram Theorem
- Gradient flows
- PDE examples
- Applications

**Books:** There will be typed lecture notes with a bibliography. For example, there will be will be material related to chapters in the following:-

H. Brezis *Functional Analysis, Sobolev Spaces and Partial Differential Equations, *Springer Universitext (2011)

A.Ern and J.-L. Guermond, *Finite Elements III, Texts in Applied Mathematics, *Springer (2021)

L. C. Evans *Partial Differential Equations,* AMS Grad Studies in Maths Vol 19

S. Bartels* Numerical Methods for Nonlinear PDEs, *Springer (2015)