Introduction to the Theory of Semigroups

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

It is known that a system of linear ordinary differential equations can be solved by virtue of the matrix exponential. However, the extension to partial differential equations is not straightforward. For example in the case of heat equation the matrix is replaced by Laplace operator which is not bounded and the series for the exponential will not converge. Moreover, solutions of the heat equation exist in general only for positive times and hence the solution operator can be at best a semigroup. The aim of the course is to provide a mathematical foundation for these types of problems and extend the concept of stability from ordinary differential equations, which is again in relation to spectrum of a linear operator.

Requirements:

Functional analysis (01FA1, 01FA2), equations of mathematical physics (01RMF), modern theory of partial differential equations (01PDR).

Syllabus of lectures:

1. Exponential of a matrix, bounded operator and possible extensions to unbounded operators.

2. Strongly continuous semigroups.

3. Uniformly continuous semigroups.

4. Analytic semigroups.

5. Semigroup generators.

6. Hille-Yoshida theorem.

7. Lumer-Phillips theorem.

8. Notions of stability.

9. Application to selected problems: relationship between spectrum and stability, exponential of unbounded operator.

Syllabus of tutorials:

Study Objective:

Knowledge: Theory of semigroups and its application to studying stability of partial differential operators (incl. the relation to spectrum).

Skills: Construction and identification of an exponential of concrete bounded and unbounded operators.

Study materials:

Povinná literatura

1. K J Engel, R Nagl, A Short: Course on Operator Semigroups, Springer, New York, 2006.

2. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

Doporučená literatura

3. L C Evans: Partial Differential Equations, 2nd ed., Amer. Mat. Soc., Providence, 2010.

4. J A Goldstein:Semigroups of Linear Operators and Applications, Second Edition, Courier Dover Publications, 2017.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství (elective course)
• Matematická informatika (elective course)