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 2025
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: English, Czech
Teaching methods: full-time
Guarantor: prof. Mgr. Milan Pokorný, Ph.D., DSc.
Teacher(s): prof. Mgr. Milan Pokorný, Ph.D., DSc.
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
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Annotation -
This is a basic course about evolutionar partial differential equations. We will deal with parabolic and linear hyperbolic equations of the second order.
Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (11.09.2013)
Course completion requirements

Exercises: the students are supposed to solve homeworks (written). To obtain credits, students must obtain at least 50% of the maximal number of points from homeworks.

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

Exam: based mostly on the material discussed at the lecture and in the written homeworks, oral exam with time to prepare the answers.

Most recommended sources are the Lecture Notes at my web page, Chapters 6-9.

Last update: Pokorný Milan, prof. Mgr., Ph.D., DSc. (25.02.2026)
Literature -

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

Skripta: https://www.karlin.mff.cuni.cz/~pokorny/LectureNotes/moderni_teorie_color.pdf

Last update: Pokorný Milan, prof. Mgr., Ph.D., DSc. (25.02.2026)
Requirements to the exam

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

The exam will be oral, with time for prepapration. The list of questions will be announced in advance.

Last update: Pokorný Milan, prof. Mgr., Ph.D., DSc. (25.02.2026)
Syllabus -

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

Nonlinear scalar elliptic equations of second order: weak formulation, uniqueness and existence theory, monotone operators, fixed point thereoms and their application.

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 and uniqueness of weak solutions.

Last update: Pokorný Milan, prof. Mgr., Ph.D., DSc. (25.02.2026)