On the kinetic $p$-Laplace equation with nonlocal diffusion

TL;DR

This paper investigates two nonlocal kinetic $p$-Laplace equations, deriving inequalities and estimates that advance understanding of nonlocal diffusion in kinetic models.

Contribution

It introduces two nonlocal models, derives representation formulas, and establishes inequalities leading to new integrability estimates for solutions.

Findings

Derived representation formulas for nonlocal kinetic models

Established homogeneous and scale-invariant inequalities

Obtained gain-of-integrability estimates for weak solutions

Abstract

We study two nonlocal versions of the kinetic $p$-Laplace equation: a Gagliardo-type model defined through differences and a Bessel-type model defined via Fourier multiplication. Using critical kinetic trajectories, we derive representation formulas adapted to the kinetic transport-diffusion geometry and establish homogeneous and scale-invariant kinetic Gagliardo-Nirenberg inequalities for nonlocal diffusion, which yield gain-of-integrability estimates for weak solutions to the kinetic $p$-Laplace equations with nonlocal diffusion.