The Harnack inequality fails for nonlocal kinetic equations

TL;DR

This paper demonstrates that the Harnack inequality, a fundamental estimate in PDE theory, does not hold for nonlocal kinetic equations, highlighting a key difference from local equations.

Contribution

It provides the first counterexample showing the failure of the Harnack inequality for nonlocal kinetic equations, specifically for the fractional Kolmogorov equation.

Findings

Harnack inequality fails for nonlocal kinetic equations

Counterexample constructed for fractional Kolmogorov equation

Harnack inequality holds for local kinetic equations like Kolmogorov

Abstract

We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.