Partial Differential Equations 2

Lecture and problem session, online, 2020

Partial Differential Equations 2. Summer semester 2020

Lecturer dr. Sebastian Schwarzacher

Assistant Claudiu Mîndrilă

EXAMINATION (added on the 18th of May 2020)

Last Wednesday at 14.15 there was the final lecture of the PDE II class. Links to the recordings will be sent via email.

The final exams we would like to do in Karlin in a large room, with you presenting at the blackbord and us asking questions from distance. I am not sure when and if everybody will be in Prague at some point. This should happen in the last week of June. A second possibility would be to schedule the exams for September. Only in case no other option is possible we can have an online exam.

Finally, we strongly recommend to use the free slots of discussions to consult with us about the final exam and what you should know there.

P.S. A reference to the parabolic maximum princples has beed added in the list with weekly recommended readings. (Week 10)

E-Lecture PDE II

Dear Students of PDE II,

we want you to be able to complete the course from the distance without too many troubles. For that we developed the following strategy:

1)The midterm is cancelled–50 Points will be distributed via homework. Another 50 Points will be distributed via the final exam.

2) The new weekly homework sheets are valid 4 points each. Every second one will be corrected by me. On each sheet we will write who to sent it to per email.

3) We offer to each of you (and we highly recommend to use this option) a weekly skype conference. The idea is that every second week you will discuss with Claudiu the homework sheets (for around 20 minutes) and in the weeks in between you will discuss with me the mathematics of the lecture (also for around 20 min). We want to keep in touch with you.

The weakly exercise sheets are below. They should be sent to or to,as it will be mentioned in the end of every sheet Please submit a .pdf file consisting of photos or typesetted in LaTeX . Write with a black pen for better visibility.


We shall refer to the following set of lecture notes.

There will be weekly assignments with instructions. Please study the subjects below and solve the related exercise sheets.

Week 1 (11-17 March)

Semigroup theory Part I

Week 2 (18-24 March)

Semigroup theory Part II

Week 3 (25-31 March)

Monotone Operators –Part I– Pages: 157–160

Week 4 (1-7 April)

Monotone Operators –Part II– Pages: 160–163

Week 5 (8-14 April)

Compact perurbations-Pages: 163-166

Week 6 (15-21 April) Fixed point methods: Pages 167-172

Week 7 (22-28 April) Aubin-Lions Theorem: Pages 218-219, 265-266 (not inluding the Proof of Thm. 7.1.3)

Week 8 “Navier-Stokes”(29 April-06 May): Pages 266-272

Week 9 Finite speed of propagation (07 May- 14 May): Pages 247-250

Week 10 Parabolic maximum principles: Pages 233-237 .

Homework sheets

  1. Please check regularly the webpage, as small typos might appear and then be corrected in the exercise sheets. (No one is perfect.) Please send any possible typos or comments to

  2. Please find HERE the situation of the points. Only the ones that chosed a nickname will appear.

Sheet 13, due on the 24th of May.

Sheet 12, due on the 13th of May. Solutions to sheet 12

Sheet 11, due on the 6th of May. Solutions to sheet 11

Sheet 10, due on the 29th of April. Solutions for sheet 10

Sheet 9, due on the 22nd of April. Solutions to sheet 9

Sheet 8, due on the 15th of April. Solutions to sheet 8

Sheet 7, due on the 8th of April. (Hint added for exercise 2) Solutions to sheet 7

Sheet 6, due on the 1st of April. (Corrected the 3rd exercise) Solutions to sheet 6

Sheet 5, due on the 25th of March. [Solutions to sheet 5]

Sheet 4, due on the 18th of March. Solutions to sheet 4

Sheet 3(corrected), due on the 13th of March. Solutions to sheet 3

Sheet 2, due on the 4th of March.

Sheet 1, due on the 26th of February.

Old rules

  1. There are 50 points available from the Homework sheets. (Not counting the bonus points)
  2. The final exam will be oral, individual, will last approx. 20 minutes . It is worth 50 points.
  3. Passing the lecture requires a minimum of 50 points.


  1. PDEs lecture notes (in Czech and English) are available here: (
  2. L. C. Evans, ‘Partial differential equations’ ; in particular the prerequisites for the current lecture can be found in the Appendix.