(in Polish) Hiperboliczne prawa zachowania

General data

Course ID:	1000-1M20HPZ
Erasmus code / ISCED:	(unknown) / (unknown)
Course title:	(unknown)
Name in Polish:	Hiperboliczne prawa zachowania
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Short description:	Brief description: The main target of this lecture is to present interesting and modern mathematical theory of hyperbolic conservation laws, deeply motivated by applications. They cover such areas as gas mechanics, traffic flow or structured population models.
Full description:	Analysis of hyperbolic conservation laws is probably one of the most difficult problems in modern theory of partial differential equations. Although they are ubiquitous in applications, their general mathematical theory is far from complete. Current results can only cover very specific cases (small initial data, scalar equation etc). We start the lecture from presenting particular models and their application. Then, we focus our attention on mathematical analysis of the models (existence of solutions, uniqueness, asymptotic behaviour). In particular, we plan to cover the following topics: 1. Existence and uniqueness for the Cauchy problem for scalar conservation laws in the class of weak entropy solutions 2. Compensated compactness methods for hyperbolic conservation laws. 3. Standard Riemann semigroup technique. If time permits, the following may be discussed during lectures or tutorials: 4. Relative entropy method and long-time asymptotics. 5. Kinetic formulation of conservation laws and applications to singular limits. 6. Conservation of energy and regularity of solutions, Onsager’s conjecture.
Bibliography:	(in Polish) Literatura: 1. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2000. 2. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, 2000. 3. B. Perthame, Kinetic Formulation of Conservation Laws, 2003. 4. C. Bardos, P. Gwiazda, A. Świerczewska-Gwiazda, E. Titi, E. Wiedemann, On the Extension of Onsager’s Conjecture for General Conservation Laws, Journal of Nonlinear Science, 29, 501-510, 2019.
Learning outcomes:	(in Polish) 1. Student zna zarys aktualnej wiedzy matematycznej dotyczącej analizy hiperbolicznych praw zachowania. 2. Zna najważniejsze problemy otwarte i rozumie trudności związane z ich rozwiązaniem. 3. Zna najważniejsze przykłady hiperbolicznych praw zachowania.
Assessment methods and assessment criteria:	(in Polish) Wykład zakończy się egzaminem ustnym. Student prezentuje wskazane zagadnienie (np. fragment artykułu naukowego). Do egzaminu w terminie zerowym dopuszczeni są wszyscy studenci.

This course is not currently offered.

Course descriptions are protected by copyright.
Copyright by University of Warsaw.