Kinetic maximal LP regularity for nonlocal Kolmogorov equation and application

Abstract

We study the linear and nonlinear nonlocal abstract Kolmogorov equations. The equations includes the abstract operator A in a Banach space E. Here,the kinetic maximal L2-regularity for the linear equat¨on is derived in terms of E-valued Sobolev spaces. Moreover, we show that the solution u is also regular in time and space variables when u is assumed to have a certain amount of regularity in velocity. Finally, the kinetic maximal L2-regularity for the linear equation can be used to obtain local existence and uniqueness of solutions to a quasilinear nonlocal Kolmogorov type kinetic equation.