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MA6L9 Variational Analysis and Evolution Equations

Lecturer: Charles Elliott

Term(s): 2

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

Assessment: Oral exam

Prerequisites: Strongly recommended to have familiarity with the material covered in MA3G7 Functional Analysis I and MA359 Measure Theory. Ideally knowledge of MA3G1 Theory of PDEs.

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)

Additional Resources