Calculus of Variations and Geometric Measure Theory

F. Anceschi - M. Piccinini

Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations

created by piccinini on 18 Jan 2023

[BibTeX]

preprint

Inserted: 18 jan 2023

Year: 2023

ArXiv: 2301.06334 PDF

Abstract:

We investigate local properties of weak solutions to nonlocal and nonlinear kinetic equations whose prototype is given by $$ \partialt u +v\cdot\nablax u +(-\Deltav)sp u = f(u). $$ 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.