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Course, academic year 2023/2024
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Partial Differential Equations 2 - NMMA406
Title: Parciální diferenciální rovnice 2
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Miroslav Bulíček, Ph.D.
Class: M Mgr. MA
M Mgr. MA > Povinné
M Mgr. MOD
M Mgr. MOD > Povinné
M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Differential Equations, Potential Theory
Annotation -
Last update: doc. RNDr. Miroslav Bulíček, Ph.D. (11.09.2013)
This is a basic course about evolutionar partial differential equations. We will deal with parabolic and linear hyperbolic equations of the second order.
Course completion requirements
Last update: prof. Mgr. Milan Pokorný, Ph.D., DSc. (07.02.2018)

Exercises: the students are supposed to solve homeworks (written). The deadlines for each set of homework will be announced at least one week in advance. To obtain credits, 30% of the maximal number of points are required. Additionally, at least one oral presentation of a problem from the list available on the web page is required. According to POS, Art. 8, Par. 2 it is not possible to repeat this.

The credit from the exercices is required to participate at the exam.

Exam: written part (based mostly on the material discussed at the exercises and contained in the written homeworks; very basic knowledge from the lectures is also expected); in order to proceed to the oral part, the students must pass the written exam

oral part (based mostly on the material presented during lectures).

Most recommended sources are the book PDE's by L.C. Evans and Lecture notes which will be available on the web page.

Literature -
Last update: doc. RNDr. Miroslav Bulíček, Ph.D. (11.09.2013)

L. C. Evans: Partial Differential Equations, AMS, 2010.

E. Zeidler: Nonlinear Functional Analysis and its Applications II/A, (Chapters 23 and 24), Springer,  1990.

Requirements to the exam
Last update: prof. Mgr. Milan Pokorný, Ph.D., DSc. (07.02.2018)

The credit from the exercises is required to to be allowed to participate at the exam.

The exam will have written part (based on problems studied at the exercises) and oral part (based on material from lectures). In order to proceed to the oral part, the student must pass the written exam. Indeed, everything is combined together and it does not mean that part of the knowledge from the exercises cannot be required at the oral exam. The exercises will be available on the webpage, the lectures are either covered by the material of the book PDE's aby L.C. Evans or by the Lecture notes. All this required theoretical material will be covered at the lectures.

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (07.01.2019)

Sobolev spaces: embedding theorems, trace theorems. (with proofs)

Nonlinear scalar elliptic equations of second order: weak formulation, uniqueness and existence theory, monotone operators, regularity, minimum and maximum principles.

Introduction to calculus of variations: fundamental theorem of calculus of variations, weak lower semicontinuity of convex functionals, relation to the elliptic equation

(Sobolev-) Bochner spaces: continuous and compact (Aubin-Lions theorem) embeddings. (with proofs)

Semigroup theory: Hille-Yosida theorem, application to linear parabolic and hyperbolic equations.

Nonlinear parabolic equations of second order: existence, uniqueness and regularity of solution.

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