On periodic homogenization of non-local convolution type operators. Part 1Лекция
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:
- construction of anzats and deriving the equation on the first corrector;
- construction of the limit diffusive operator;
- 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.