### Evolution Equations (Summer Semester 2020)

- Lecturer: Prof. Dr. Roland Schnaubelt
- Classes: Lecture (0156400), Problem class (0156410)
- Weekly hours: 4+2
- Audience: Mathematics (from 6. semester)

** Lectures and exercises will be held online (at first). I will upload the text on board and my oral explanations as a video file (mp4) in ILIAS. In addition I want to offer online discussions via Microsoft Teams. Details can be found in Ilias.
**

**The lectures are based on my manuscript from winter 2018/19, which can be found on my webpage. I will upload a revised in Ilias parallel to the lectures.**

Schedule | |||
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Lecture: | Monday 9:45-11:15 | SR 3.061 | Begin: 20.4.2020 |

Wednesday 8:00-9:30 | SR 3.068 | ||

Problem class: | Friday 9:45-11:15 | SR 3.068 | Begin: 24.4.2020 |

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

Office hours: Monday at 10:00 - 11:00, and by appointment. | ||

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

Email: schnaubelt@kit.edu | ||

Problem classes | Dr. Nick Lindemulder | |

Office hours: | ||

Room 2.044 Kollegiengebäude Mathematik (20.30) | ||

Email: nick.lindemulder@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 can be applied to the diffusion, the (damped) wave, and the Schrödinger equation.

It is stronlgy 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) ansd discussed.

# Examination

There will be an oral exam (of about 30 min) on **17.8.** or **25.9.** Please register online at CAS Campus. Afterwards select one of the examination days in an email to schnaubelt@kit.edu (**up to 12.8. for the first date, up to 22.9. for the second**). You will then be informed about the time of the exam.

# References

On my webpage one can find the PDF file of the manuscript of my lecture Evolution Equations from winter semester 2018/19. It will be updated during this semester.

The following monographs on evolution equations can be found in the department's library. (You have online access to Engel/Nagel via the KIT library.)

- Engel, Nagel: One-Parameter Semigroups for Linear Evolution Equations
- Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations

- Arendt, Batty, Hieber, Neubrander: Vector-valued Laplace Transforms and Cauchy Problems
- Davies: One-Parameter Semigroups
- Engel, Nagel: A Short Course of Operator Semigroups
- Fattorini: The Cauchy Problem
- Goldstein: Semigroups of Linear Operators and Applications
- Hille, Phillips: Functional Analysis and Semi-groups
- Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems
- Tanabe: Equations of Evolution