Winter 2015/16

BMS-Course Infinite-Dimensional Dynamics

Prof. Dr. Bernold Fiedler

Recitation session: Isabelle Schneider

Schedule, Winter 2015/16

Lecture:

Tuesday, 10.15-14.00, seminar room 130, Arnimallee 3 (rear building)

Recitation session:

Thursday, 10.15-12.00, seminar room 130, Arnimallee 3 (rear building)

Checking the written examination: March 9, 3 p.m.

Written examination:

Tuesday, February 9, 2016, 10:15-11:45

(Results)

Resit:

Wednesday, April 20, 10.15-11:45

(Results)

Topics

Infinite-dimensional dynamics today mainly addresses partial differential equations and delay differential equations, from a dynamical systems perspective. Technical prerequisites like semigroup theory and invariant manifolds will be surveyed in class.

The aim is to arrive at current concepts of global attractors, which capture the long-time dynamics. This is not limited to just one or the other single solution, but aims at their global geometry. Relevant topics include finite-dimensionality, determining nodes and modes, inertial manifolds, homogenization, delay control, and others, depending on the preferences of the audience.

Thesis work will develop in class.

Prerequisites

Basic concepts of dynamical systems OR basic concepts of partial differential equation - and a fresh mind.

References

• H.-W. Alt: Lineare Funktionalanalysis. Springer, 1985.
• R. Courant and D. Hilbert: Methoden der Mathematischen Physik I, II. Springer, 1924.
• L.C. Evans: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, 1998.
• A. Friedman: Partial Differential Equations of Parabolic Type. Prentice Hall, 1964.
• A. Friedman: Partial Differential Equations. Holt et al., 1969.
• D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, 1977.
• D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, 1981.
• T. Kato: Perturbation Theory for Linear Operators. Springer, 1966.
• A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, 1983.
• H. Tanabe: Equations of Evolution. Pitman, 1979.
• E. Zeidler: Vorlesungen über nichtlineare Funktionalanalysis I-V. Teubner, 1980-1982. English translation by Springer, 1988-1993.

see for a general background:

• D. Henry: Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981.
• A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
• H. Tanabe, Equations of Evolution, Pitman, Boston, 1979.

see for global attractors in general:

• A.V. Babin and M.I. Vishik: Attractors of Evolution Equations, North Holland, Amsterdam, 1992.
• V.V. Chepyzhov and M. I. Vishik: Attractors for Equations of Mathematical Physics, Colloq. AMS, Providence, 2002.
• A. Eden, C. Foias, B. Nicolaenko and R. Temam: Exponential Attractors for Dissipative Evolution Equations, Wiley, Chichester, 1994.
• J.K. Hale: Asymptotic Behavior of Dissipative Systems, Math. Surv., 25. AMS Publications, Providence, 1988.
• J.K. Hale, L. T. Magalhaes and W. M. Oliva, Dynamics in Infinite Dimensions, Springer- Verlag, New York, 2002.
• O.A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.
• G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, 2 (2002), 885-982.
• G. R. Sell and Y. You: Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
• R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer- Verlag, New York, 1988.

see for functional differential equations:

• R. Bellman and K.L. Cooke: Differential-Difference Equations. Academic Press, New York,1963.
• O. Diekmann, S.A. van Gils, S. M. Verduyn-Lunel and H. O. Walther: Delay Equations: Functional-, Complex-, and Nonlinear Analysis, vol. 110. Springer, New York, 1995.
• J.K. Hale: Theory of Functional Differential Equations. Springer, New York, 1977.
• J.K. Hale and S.M. Verduyn-Lunel: Introduction to Functional Differential Equations. Springer, New York, 1993.
• V. Kolmanovski and A. Myshkis: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer, Dordrecht, 1999.
• R.G. Nussbaum: Functional differential equations. In: B. Fiedler (ed.): Handbook of Dynamical Systems, vol. 2, pp. 461-499. Elsevier/North-Holland, Amsterdam, 2002.
• J. Wu: Theory and Applications of Partial Functional Differential Equations. Springer, New York, 1996.

Homework assignments, Winter 2015/16

1. assignment, due October 26, 2015 (PDF)
2. assignment, due November 2, 2015 (PDF)
3. assignment, due November 9, 2015 (PDF)
4. assignment, due November 16, 2015 (PDF)
5. assignment, due November 23, 2015 (PDF)
6. assignment, due November 30, 2015 (PDF)
7. assignment, due December 7, 2015 (PDF)
8. assignment, due December 14, 2015 (PDF)
9. voluntary assignment for the Christmas holidays, due January 11, 2016 (PDF)
10. assignment, due January 18, 2016 (PDF)
11. assignment, due January 25, 2016 (PDF)
12. assignment, due February 1, 2016 (PDF)

Core questions, Winter 2015/16

• (Questions 1)
• (Questions 2)