ISEM30: Evolution equations governed by accretive operators
Short description: The 30th International Internet Seminar 2026/27 deals with (in general nonlinear) abstract Cauchy problems of the form u'+Au∋f on [0,T], u(0)=u_0, where A is an accretive operator on a Banach space X. Many partial differential equations in science can be abstractly rewritten in this way, and a theory for the abstract Cauchy problem allows to treat them in a unified way. The following topics will be treated:
1) Accretive operators and subgradients
2) Wellposedness of the abstract Cauchy problem: existence, uniqueness and continuous dependence of solutions on data (f and u_0)
3) Approximation theory
4) Qualitative theory: e.g. comparison principles, regularity of solutions, ...
5) Asymptotic behaviour
6) Applications like diffusion equations, transport equations, phase separation prolems, wave phenomena, ...
The title of this years internet seminar is the title of an unpublished book on the topic by Philippe Benilan, Michael Crandall and Amnon Pazy. This book was already cited in Haim Brezis' book Analyse fonctionnelle from 1983 as "to appear", but unfortunately it has never appeared. This internet seminar is also meant to honour this unpublished book and the mathematicians behind.
Virtual lecturer: Ralph Chill (TU Dresden)
Registration on OPAL for access to the lectures and a discussion forum. Registration is free. Register first on OPAL by following the option "You do not have an email address at a Saxon university", choosing a login and a password, and choosing the option "TU Dresden". Once you are registered, then sign up for the OPAL course.
The internet seminar is temporarily divided into three phases:
Lecture phase: October 2026 - February 2027
Project phase: February 2027 - May 2027
Final workshop: 24-28 May 2027
Lecture phase: During the lecture phase, the "virtual lecturer" uploads a new chapter of lecture notes on OPAL each week. Each chapter consists of approximately 10 pages plus a collection of exercise problems. In addition to the lecture notes, there will be an asynchronous video lecture. The participants read the lecture notes, watch the videos (if they like, not necessary) and work on the exercise problems. While doing so, they discuss and collaborate in two different ways:
- Via the forum on OPAL, which has LaTeX support.
- Via local groups, seminars or courses at the participants' universities. At many universities, the participating students or lecturers organize a local ISEM group that meets regularly to discuss the lectures and the exercise problems.
Each week, the virtual lecturers will ask the local group of one university to provide the solutions to the week's exercises written in LaTeX, such that they can be uploaded to the ISEM webpage.
Project phase: Students who are interested to get even more engaged after the lecture phase can apply to participate in the ISEM's project phase. The participants of the project phase will be split into international groups of 3-5 students coordinated by experienced mathematicians. Each group studies one specific topic in depth - usually by reading a research article about the topic and discussing it online. Each group prepares a presentation for the on-site workshop in the third and final phase of the ISEM. Thus, participating in the project phase implies a commitment to participate in person in the final workshop!!
Workshop: The final workshop takes place at Kloster Marienthal / Ostritz from 24 to 28 May 2027 (arrival on 23 May, departure on 28 May).
The Internet Seminars: The "Internet Seminar on Evolution Equations" is an international mathematical event that is organized annually by different mathematicians. Each year's organizers choose a topic - usually closely related to evolution equations and/or operator theory - and design a course that consists of three phases: an online lecture phase, an online project phase and an on-site workshop. Students from all over the world participate in the event.