Semigroups and Evolution Equations (V5B2) -- Part I

• No oral exams on March 16/17/18/19 due to the Corona virus! More information on new appointments will follow as soon as possible.
• Monday 12-14, room MATH / SR 0.011 
• Lecture notes
• Oral exams (the appointments for the oral exams will be made in class on January, 20)
  • February 10/11/12/13
  • March 16/17/18/19
• There are two corrections in the lecture notes. An additional assumption has been added in Lemma 2.32 and Theorem 4.7 (ii).

 

Content of the lecture:

Important questions in the study of PDEs are those for existence, uniqueness and qualitative behaviour of solutions. In this lecture the focus is on time-dependent PDEs and the theory of semigroups as a tool to obtain answers for these questions. The main idea of semigroup theory in this context is to interpret a given PDE as an abstract ODE in an infinite dimensional Banach space. We will see that solutions of linear evolution equations are constructed by a semigroup which may be seen as a generalisation of the matrix exponential for unbounded operators on Banach spaces.

The aim of this lecture is to give an insight into parts of the basic theory of semigroups and how this theory may be used to solve abstract evolution equations. We first introduce some fundamental results on (strongly continuous) semigroups and their infinitesimal generators. Then the lecture is concerned with so-called strongly continuous semigroups and their application to hyperbolic PDEs.

 

In the summer term 2020 there will be a second part of this lecture in which we treat analytic semigroups and their application to parabolic PDEs.

 

 

Prerequisites:

Basic notions of functional analysis and PDEs should be known.