Submitted Paper

Inserted: 16 jul 2024
Last Updated: 14 aug 2024

Year: 2024
Links: link to ResearchGate

Abstract:

We deal with the following wide class of kinetic equations, \[ \partial_t f \,+\, v\cdot\nabla_x f \, = \, \mathcal{L}_v f. \] Above, the diffusion term $\mathcal{L}_v$ is an integro-differential operator, whose nonnegative kernel is of fractional order $s\in(0,1)$ having merely measurable coefficients.

Firstly, we obtain a general $L^\infty$-interpolation inequality with a natural nonlocal tail term in velocity, in turn giving local boundedness even for weak subsolutions $f$ without any sign assumptions. This is a veritable novelty, being boundedness usually assumed a priori in such a setting. Then, provided that their nonlocal tail in velocity is $(2+\varepsilon)$-summable along {the transport variables}, we prove a general Strong Harnack inequality, which in the simplest case of globally nonnegative weak solutions $f$ reads as follows \[ \sup_{Q^-} f \ \leq \ c\inf_{Q^+} f \,+ \, c\, |
{Tail} (f) |
_{L^{2+\varepsilon}_{t,x}} \] where $Q^{\pm}$ are suitable slanted cylinders. This is the first Strong Harnack inequality for kinetic integro-differential-type equations under the aforementioned tail summability assumption, which is in fact naturally implied in literature, e. g., from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and in clear accordance with the very recent surprising counterexample by Kassmann and Weidner; see arxiv:2405.05223

A new standalone result, a Besicovitch-type covering argument for very general kinetic geometries, independent on the involved equation, is also needed, stated and proved.

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• Anceschi-Palatucci-Piccinini_Harnack-inequalities-for-kinetic-integral-equations.pdf