Preprint

Inserted: 18 jan 2023
Last Updated: 5 apr 2024

Year: 2023

Abstract:

We investigate local properties of weak solutions to nonlocal and nonlinear kinetic equations whose prototype is given by $ \partial_t u +v\cdot\nabla_x u +(-\Delta_v)^s_p u = f(u) $. We consider equations whose diffusion part is a (possibly degenerate) integro-differential operator of differentiability order $s \in (0,1)$ and summability exponent $p\in (1,\infty)$. Amongst other results, we provide an explicit local boundedness estimate by combining together a suitable Sobolev embedding theorem and a fractional Caccioppoli-type inequality with tail. For this, we introduce in the kinetic framework a new definition of nonlocal tail of a function and of its related tail spaces, also by establishing some useful estimates for the tail of weak solutions. Armed with the aforementioned results we give a precise control of the long-range interactions arising from the nonlocal behaviour of the involved diffusion operator.

Keywords: fractional Laplacian, fractional Sobolev spaces, kinetic equations, Kolmogorov-Fokker-Planck equations, boundedness estimates, nonlinear operators