(in Polish) Ewolucyjne równania różniczkowe cząstkowe. Przegląd podstawowych metod ich badania 1000-1M20ERR
The course is devoted to the presentation of a selection of methods for
studying evolutionary problems. We shall regard them as infinitely
dimensional dynamical system. A particularly nice example is the
reaction-diffusion equations.
It is well-known from the ODE theory that investigating stability of the
steady states and orbits connecting them is important. A particular
example of such a solution to reaction-diffusion problems are traveling
waves.
Another important example are self-similar solutions. They appear in
different types of nonlinear problems. They are the key ingredients in
the construction of shock waves. This is a type of solution to
hyperbolic conservation laws. These problems are completely different
from the diffusion-reaction equations.
The structure of an equation simplifies if we know that it is a
gradient flow or at least it has a Lapunov functional. One of important
problems of this sort is the Cahn-Hilliard equation, which is of fourth
order. Due to the the Lapunov functional we can show existence for all
positive times.
The lecture is for students interested in PDEs, no prerequisite is
required beyond the basic PDE course. More topics than mentioned above
will be covered.
Main fields of studies for MISMaP
Type of course
Mode
Self-reading
Learning outcomes
A Student:
1. knows the importance of the study of stability of the steady states
2. knows importance of the travelling waves and self-similar solutions for the study of the dynamics
3. knows examples of the gradient flows and knows the notion of the omega-limit sets.
Assessment criteria
A student is supposed to write an essay on a topic related to the course. A final grade is issued after the conversation on this essay.
Bibliography
Christian Kuehn, PDE dynamics. An introduction. Mathematical Modeling
and Computation, 23. Society for Industrial and Applied Mathematics
(SIAM), Philadelphia, PA, 2019
Alain Miranville, The Cahn-Hilliard equation. Recent advances and
applications. CBMS-NSF Regional Conference Series in Applied
Mathematics, 95. Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 2019
Guido Schneider, Hannes Uecker, Nonlinear PDEs: A Dynamical Systems
Approach, AMS, Providence, RI, 2017
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: