### Evolution Equations (Winter Semester 2023/24)

- Lecturer: Prof. Dr. Roland Schnaubelt
- Classes: Lecture (0105900), Problem class (0105910)
- Weekly hours: 4+2
- Audience: Mathematics (from 7. semester)

**All information and materials for this lecture are provided in the course "Evolution Equations" within Ilias, including communication via email and forum. Please join this course if you want to participate.**

Schedule | |||
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Lecture: | Tuesday 9:45-11:15 | SR 2.067 | Begin: 24.10.2023 |

Thursday 15:45-17:15 | SR 3.069 | ||

Problem class: | Monday 15:45-17:15 | SR 3.069 | Begin: 23.10.2024 |

Lecturers | ||
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Lecturer | Prof. Dr. Roland Schnaubelt | |

Office hours: Tuesday at 12:00 - 13:00, and by appointment. | ||

Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30) | ||

Email: schnaubelt@kit.edu | ||

Problem classes | Richard Nutt M.Sc. | |

Office hours: Just drop by whenever I am in the office. | ||

Room 2.043 Kollegiengebäude Mathematik (20.30) | ||

Email: richard.nutt@kit.edu |

Evolution equations describe the time evolution of dynamical systems by an ordinary differential equation in a Banach space. We investigate linear and autonomous (time-invariant) problems. In this case the solutions are given by a one-parameter semigroup of linear operators. For such operator semigroups there is a quite complete theory, which allows us to study the properties of the underlying dynamical system. This approach essentially relies on functional analytic methods and results.

We treat the basic existence theorems for linear autonomous evolution equations. In this framework, we then investigate qualitative properties of the solutions, such as regularity and the longterm behavior. Perturbation and approximation results are also studied (which have connections to numerical analysis). The developed theory will be applied to the diffusion, the (damped) wave, and the Schrödinger equation, for instance.

It is strongly recommended to have attended a lecture in functional analysis and on the theory of L^p spaces. The necessary parts from the lecture Spectral Theory will be recalled (without proofs) and discussed.

# Examination

There will be an oral exam (of about 30 min) on **5 March** or on **27 March** starting from 9:00. It takes place in room 2.070. Please register online at CAS Campus Management. Afterwards come to the secretariat Katz/Schaaf 2.041 to select one oft the dates (until 29 February (12:00) for the first, until 22 March (12:00) for the second). You can deregister from the exam (without giving a reason) up to 3 working days before the exam. If you have obtained a time slot from the secretariat, please inform it and me about your withdrawal. The exam can be taken in Englisch or German.

You will not be asked about the additional contents of the lecture notes (Theorems 4.19-21, Chapter 5, proofs omitted in the lectures (marked by a footnote), and references to the literature and other lecture notes). However, you should understand why a cited result can be used in the arguments.

# References

On my webpage and in Ilias, parallel to the lectures I will provide lecture notes. Here are several relevant monographs, including classics (the KIT library provides online access to Engel/Nagel):

- K.J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Springer, 2000.
- A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, 1983

- W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, 2011.
- E.B. Davies: One Parameter Semigroups. Academic Press, 1980.
- K.J. Engel, R. Nagel: A Short Course of Operator Semigroups. Springer, 2006.
- H.O. Fattorini: The Cauchy Problem. Addison-Wesley, 1983
- J.A. Goldstein: Semigroups of Linear Operators and Applications. Oxford University Press, 1985.
- E. Hille, R.S. Phillips: Functional Analysis and Semigroups. American Mathematical Society, 1957.
- A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, 1995.
- H. Tanabe: Equations of Evolution. Pitman, 1979.