11–18 June 2021
These are mini-courses for the EIMI thematic program on Spectral Theory and Mathematical Physics (rescheduled from 2020 due to COVID-19 pandemic). The target audience includes graduate, master and senior bachelor students of any mathematical speciality; senior researchers are also welcome.
The meetings will be held in the Leonhard Euler International Mathematical Institute (Pesochnaya nab. 10, St. Petersburg, Russia). The talks will be recorded and posted on the event web-page.
Please, see the timetable page for the schedule.
11–18 June, 2021
The course will focus on periodic homogenization of parabolic and elliptic problems for integral convolution type operators, it is based on recent results obtained in our joint works with A.Piatnitski.
First, we will consider some models of population dynamics, which can be described in terms of non-local convolution type operators. In particular, we will introduce evolution equations for the dynamics of the first correlation function (the so-called density of population). We will also discuss a number of problems arising in the theory of non-local convolution type operators.
Then we will turn to homogenization of elliptic equations for non-local operators with a symmetric kernel. We will show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. The proof of convergence includes the following steps:
Our next goal is homogenization of parabolic problems for operators with a non-symmetric kernel. It will be shown that the homogenization result holds in moving coordinates. We will find the corresponding effective velocity and obtain the limit operator. In the case of small antisymmetric perturbations of a symmetric kernel we will show that the so-called Einstein relation holds.
16–18 June, 2021
We discuss long time asymptotics of the solution to a Cauchy problem for the nonlinear cylindrical KdF equation. Our method is based on an asymptotic analysis of a matrix Riemann-Hilbert problem. We use a well-known version of the nonlinear steepest descent method.