Course: Nonlinear Continuum Mechanics

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Course title Nonlinear Continuum Mechanics
Course code KMA/NLUM
Organizational form of instruction Lecture + Seminar
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 4
Language of instruction Czech
Status of course unspecified
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Fürst Tomáš, RNDr. Ph.D.
  • Horák Jiří, doc. RNDr. CSc.
  • Netuka Horymír, RNDr. Ph.D.
  • Ligurský Tomáš, RNDr. Ph.D.
Course content
A. The language of dynamical systems 1. A summary of linear systems 2. De-motivation: Chaos and the Lorenz attractor 3. Case-study: Driven damped oscillations 4. Relation of non-linear to linear systems: The Hartman-Grobman Theorem 5. Chaos forbidden: The Poincare-Bendixon Theorem B. Fixed-point methods 1. Fixed-point theorems 2. Standard applications 3. BVP for nonlinear ODEs 4. nonlinear PDEs: Classical approach 5. nonlinear PDEs: Modern approach 6. modern approach to nonlinear evolution PDEs. C. Monotonicity methods 1. Monotonicity and the Browder-Minty Theorem 2. Application: The method of lower and upper solutions 3. Pseudo-monotonicity and the Brezis Theorem 4. Application to steady state problems 5. Application to evolution problems

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration
  • Preparation for the Exam - 45 hours per semester
  • Attendace - 65 hours per semester
  • Semestral Work - 15 hours per semester
Learning outcomes
Understand the language of dynamical systems and solution methods for nonlinear ODEs and PDEs based on fixed-point principles and monotonicity.
Comprehension Understand the mathematical tools of nonlinear partial differential equations.
Classical and modern theory of ODEs and PDEs, Lebesgue's theory, calculus.
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Assessment methods and criteria
Oral exam, Written exam

seminar work
Recommended literature
  • C. Robinson. (1999). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press.
  • E. Feireisl. (2004). Dynamics of viscous compressible fluids. Oxford: Oxford University Press.
  • J. Lukeš. (2001). Zápisky z funkcionální analýzy. MatFyzPress.
  • L.C. Evans. (1998). Partial differential equations. AMS.
  • P. Drábek, J. Milota. (2004). Lectures on Nonlinear Analysis. Plzeň.
  • Roubíček T. (2008). Nonlinear Partial Differential Equations with Applications. Birkhauser.
  • V. M. Aleksejev, V. M. Tichomirov, S. V. Fomin. (1991). Matematická teorie optimálních procesů. Academia, Praha.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester