Offline Mini-Courses in Spectral Theory and Mathematical Physics

Europe/Moscow
Leonhard Euler International Mathematical Institute in St. Petersburg

Leonhard Euler International Mathematical Institute in St. Petersburg

St. Petersburg, Pesochnaya nab. 10, 197022, Russia
Description

Offline Mini-Courses
in Spectral Theory and Mathematical Physics

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.

Elena Zhizhina (Institute for Information Transmission Problems, Russia)
On periodic homogenization of non-local convolution type operators

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:

  1. construction of anzats and deriving the equation on the first corrector;
  2. construction of the limit diffusive operator;
  3. justification of the convergence.

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.

References:

  1. A.Piatnitski, E.Zhizhina, Periodic homogenization of non-local operators with a convolution type kernel, SIAM J. Math. Anal. Vol. 49, No. 1, p. 64-81, 2017.
  2. A.Piatnitski, E.Zhizhina, Homogenization of  biased convolution type operators, Asymptotic Analysis, 2019, Vol. 115, No. 3-4, p. 241-262, doi:10.3233/ASY-191533; Arxiv: 1812.00027.
  3. Yu.Kondratiev, O.Kutoviy, S.Pirogov, E.Zhizhina, Invariant measures for spatial contact model in small dimensions, Arxiv: 1812.00795, 29 November 2018, Markov Proc. Rel. Fields (2021), to appear.

Alexander Its (IUPUI, USA & St. Petersburg University, Russia)
Nonlinear saddle point method for the cylindrical KdF equation

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.

    • 10:00 10:45
      On periodic homogenization of non-local convolution type operators (1/6) 45m
      Speaker: Elena Zhizhina
    • 10:45 10:50
      Break 5m
    • 10:50 11:35
      On periodic homogenization of non-local convolution type operators (2/6) 45m
      Speaker: Elena Zhizhina
    • 11:35 12:00
      Coffee break 25m
    • 10:00 10:45
      Nonlinear saddle point method for the cylindrical KdF equation (1/4) 45m
      Speaker: Alexander Its
    • 10:45 10:50
      Break 5m
    • 10:50 11:35
      Nonlinear saddle point method for the cylindrical KdF equation (2/4) 45m
      Speaker: Alexander Its
    • 11:35 12:00
      Coffee break 25m
    • 12:00 12:45
      On periodic homogenization of non-local convolution type operators (3/6) 45m
      Speaker: Elena Zhizhina
    • 12:45 12:50
      Break 5m
    • 12:50 13:35
      On periodic homogenization of non-local convolution type operators (4/6) 45m
      Speaker: Elena Zhizhina
    • 10:00 10:45
      Nonlinear saddle point method for the cylindrical KdF equation (3/4) 45m
      Speaker: Alexander Its
    • 10:45 10:50
      Break 5m
    • 10:50 11:35
      Nonlinear saddle point method for the cylindrical KdF equation (4/4) 45m
      Speaker: Alexander Its
    • 11:35 12:00
      Coffee break 25m
    • 12:00 12:45
      On periodic homogenization of non-local convolution type operators (5/6) 45m
      Speaker: Elena Zhizhina
    • 12:45 12:50
      Break 5m
    • 12:50 13:35
      On periodic homogenization of non-local convolution type operators (6/6) 45m
      Speaker: Elena Zhizhina