Damir Kinzebulatov, Université Laval, Canada
I will discuss various realizations of Nash's method that allows us to study heat kernels of local and non-local Kolmogorov equations with singular drifts. The latter includes some drifts arising in particle systems with strong attracting interactions of Keller-Segel type. Crucially, we can treat the strength of attraction going almost up to the blow-up threshold. One way to handle such drifts, even in the non-local situation, is to introduce certain desingularizing weights and run Nash's method in a weighted space. There is a certain analogy with Carleman's method of proving unique continuation that uses similar singular weights. In the last part of my talk I will discuss how this analogy can be exploited.
The talk is based on some joint papers with S.E.Boutiah, R.Gibara, K.R.Madou, Yu.A.Semenov and K.Szczypkowski.