Nonlinear Dynamics at the Free University Berlin

Winter 2016/17

# Differentialgleichungen II

PD Dr. Martin Väth

Exercises: Nicola Vassena

## Schedule, Winter 2016/17

Lecture:
Monday 14.15-16.00, SR 032, Arnimallee 6
Wednesday, 10.15-12.00, SR E.31, Arnimallee 7
Exercises:
Monday, 16.00-17.45, SR 032, Arnimallee 6
On the first day of the semester (Monday, 17.10.2016), instead of lecture and exercise there will be the Nachklausur (repetition of the final exam) of Differentialgleichungen I

Dates to look up the Exam: October 19-November 2, 2016, Room 136, Arnimallee 7. If I should not be there, ask in the office 142 (or better fix a date with me e.g. by email).

## Registration

Please register (in addition to the standard registration) into KVV. This is not a substitute for the standard registration and has no obligations; for instance, you need not be afraid to register into KVV if you are not sure yet whether you will join the course, but it will help us in organizing things.

Instructions to do this:

1. Go to the KVV homepage.
2. Login with Direkter KVV-Login with your ZEDAT account. (If you do not have a ZEDAT account and cannot easily get one, please ask after lecture or exercises.)
3. Go to Membership, choose Joinable Sites, choose the lecture (Basismodul: Differentialgleichungen II) and press the join button.

## Content

The plan is to divide the lecture roughly into 3 parts:

1. A primer on advanced techniques for dynamical systems (Hartman-Grobman theorem, Ljapunov function techniques, stable and unstable manifolds, a sketch on center manifolds and how they are applied)
2. A primer on PDEs (the crucial 3 classes of PDEs - elliptic, parabolic, hyperbolic - with their main examples Laplace, heat, and wave equation), followed by functional analytical techniques (Sobolev spaces together with their main embedding theorems and inequalities, L2 theory for elliptic equations)
3. Semilinear parabolic and hyperbolic PDEs: Semigroup theory (for analytic and C0 semigroups, respectively) and the corresponding functional analytic background for unbounded operators.

This is a very huge program, and presumably either from the second or third part some topics (or at least some proofs) have to be omitted: It will depend on the audience what will be considered in more detail. Since the presumed main audience will be from Differentialgleichungen I, presumably no pre-knowledge will be expected in functional analysis or PDEs.

For the first part, knowledge in basic differential equations is required, and hopefully the concept of a flow is not completely new.

For the other parts, knowledge in Lebesgue integration theory is recommended, although probably a small sketch of its main concepts will be recalled.

## Literature

For the first part (in decreasing order of relevance for the lecture):
1. H. Amann, Ordinary differential equations, Walter de Gruyter, 1990
2. G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012
3. J. W. Prüss and M. Wilke, Gewöhnliche Differentialgleichungen und Dynamische Systeme, Birkäuser, 2010
4. P. Hartman, Ordinary Differential Equations, Cambridge University Press, 2002
5. A. Vanderbauwhede, Center Manifolds, Normal Forms and Elementary Bifurcations, In: Dynamics Reported, Vol. 2, Wiley, 1989
For the second (and partly third) part:
1. H. Brezis, Functional analysis, Sobolev spaces, and partial differential equations, Springer, New York, Dordrecht, Heidelberg, London, 2011
2. L. C. Evans, Partial differential equations, 2nd ed., Amer. Math. Soc., Providence, R. I., 2010
3. H. W. Alt, Lineare Funktionalanalysis, 2nd ed., Springer, Berlin, Heidelberg, New York, 1992
4. L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford, 1982
5. J. Appell and M. Väth, Elemente der Funktionalanalysis, Vieweg & Sohn, Braunschweig, Wiesbaden, 2005
6. M. Väth, Integration theory. A second course, World Scientific Publ., Singapore, New Jersey, London, Hong Kong, 2002
7. T. Kato, Perturbation theory for linear operators, Springer, New York, 1966
8. D. Werner, Funktionalanalysis, 4th ed., Springer, Berlin, Heidelberg, New York, 2002
9. N. Dunford and J. T. Schwartz, Linear operators I, 3rd ed., Int. Publ., New York, 1966
10. H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, New York, Oxford, 1978
For the third (and partly second) part:
1. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, Berlin, Heidelberg, 1992
2. D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math. 840, Springer, Berlin, New York, 1981
3. H. Amann, Linear and quasilinear parabolic problems I , Birkhäuser, Basel, Boston, Berlin, 1995
4. A. Belleni-Morante and A. C. McBride, Applied nonlinear semigroups, John Wiley & Sons, Chichester, New York, Weinheim, 1998
5. A. Lunardi, Analytic semigoups and optimal regularity in parabolic problems, Birkhäuser, Basel, Boston, Berlin, 1994
6. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ., Providence, R. I., 1957
7. I. I. Vrabie, C0-semigroups and applications, 2nd ed., Elsevier, Amsterdam, 2003
8. E. M. Ouhabaz, Analysis of heat equations on domains, Princeton Univ. Press, Princeton, Oxford, 2005
Most of these books are in the library in the "Handapparat".

## Dates for the Exam (Klausur)

The Klausur will be on February 8, 2017 during lecture.
Start of the Klausur already on 10:00

You can bring one (A4) paper with you for the exam on which you can have written whatever you want. Please also bring empty paper and a pen with you! Nothing else is admissible (in particular: no calculators, computers, smartphones, books, lecture notes etc.)

## Exercises for winter 2016/17 and their finishing dates

For attending the final exam, every person needs to obtain 50% of the total points of the exercises and, in addition, must regularly have presented solutions to the exercises in the tutorials. ("Regularly" means at the very least 1-3 times, depending on the total number of exercises and how they are split, and on the number of attendants.) Please form groups of not more than three people and hand in your joint solutions. Please note clearly your name and Matrikelnummer on your solutions. Please staple your solutions together if you hand in multiple pages. The last date in which you can hand in is Wednesday 20:00; you can put your solutions into Nicola Vassena's box (Tutorenfächer) in Arnimallee 3, ground floor (just beside the stairs).

Please write the solutions to the exercises in English (because they are corrected by Nicola)!

1. exercise01.pdf October 26, 2016
2. exercise02.pdf November 2, 2016
3. exercise03.pdf November 9, 2016
4. exercise04.pdf November 16, 2016
5. exercise05.pdf November 23, 2016
6. exercise06.pdf November 30, 2016
7. exercise07.pdf December 7, 2016
8. exercise08.pdf December 14, 2016
9. exercise09.pdf January 9, 2017
10. exercise10.pdf January 11, 2017
11. exercise11.pdf January 18, 2017
12. exercise12.pdf January 25, 2017
13. exercise13.pdf February 1, 2017

## Suggestions for some solutions of current exercises

The solutions will be online only for 1-2 weeks after the finishing dates.
1. solution01.pdf
2. solution02.pdf
3. solution03.pdf
4. solution04.pdf
5. solution06.pdf
6. solution07.pdf
7. solution08.pdf
8. solution09.pdf
9. solution10.pdf
10. solution11.pdf
11. solution12.pdf
12. solution13.pdf
13. circle.pdf
14. (Characterization of 1-dimensional manifolds)

## Manuscript

The current manuscript will be online only for 1-2 weeks after the corresponding lecture.

For part 1, the manuscript is in German, the later parts will be in English.
1. dgl1.pdf
2. dgl2a_01.pdf
3. dgl2a_02.pdf
4. dgl2b_01.pdf
5. dgl2b_02.pdf
6. dgl2b_03.pdf
7. dgl2b_04.pdf
8. dgl2b_05.pdf New version on January 23: Fix wrong signs in weak formulation of Neumann case and typos in coercivity estimate (p. 38). Formulate Gagliardo-Nirenberg slightly more general.
9. dgl2b_06.pdf (Proof of main embedding theorems)
10. dgl2b_07.pdf (C0 semigroups, part 1|2)
11. dgl2b_08.pdf (C0 semigroups, part 2|2)
Last change: Feb. 6, 2017
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