Partial Differential Equations [Fall 2019]
A basic knowledge of real and complex analysis and/or calculus. A background in ordinary differential equations and/or dynamical systems.
Aim of the course
The aim of the course is to gain an understanding of some of the basic techniques that underpin modern research in the field of partial differential equations.
The course is taught bij Hermen Jan Hupkes (HJH) and Joost Hulshof (JH).
HJH's part of the course can be split into two portions. In the introductory portion, he discusses how PDEs differ from ODEs and what the natural questions are that can be considered, using the transport equation, the Laplace equation and Poisson's equation on bounded and unbounded domains as guiding examples. We will loosely follow Chapters 1 and 2 of Evans' book Partial Differential Equations (second edition), but also discuss Green's identities, adjoint problems and Fourier analysis.
In the second portion we study semigroups, based on Evans and Engel&Nagel. We discuss several different notions of continuity (uniform, strong and maybe even analytic) and prove the Hille-Yosida generation theorem. A great deal of time will be spent on examples to illustrate how the abstract results can be applied to concrete PDEs.
In JH's part of the course he will do Chapters 5 and 6 of Evans' book Partial Differential Equations (second edition) and parts of the appendices needed tot this purpose. The goal is to establish the basic theory for (linear) elliptic boundary value problems via the functional analytic approach using weak (re-)formulation in suitable Sobolev spaces, which will be treated in considerable detail, as well as the regularity of solutions needed for the semigroup approach to linear evolution problems. The fundamental role of the integral inequalities will also be illustrated with some basic observations concerning the Navier-Stokes equations.
J. Hulshof (VU) & H-J. Hupkes (UL)