Submitted Paper

Inserted: 13 jan 2024
Last Updated: 5 apr 2024

Pages: 41
Year: 2024

41pp 4 figure

Abstract:

We deal with a wide class of kinetic equations, \[ \big[ \partial_t + v\cdot\nabla_x\big] f = \mathcal{L}_v f. \]

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. Amongst other results, we are able to prove that nonnegative weak solutions $f$ do satisfy \[ \sup_{Q^-} f \ \leq \ c\inf_{Q^+} f, \] where $Q^{\pm}$ are suitable slanted cylinders. No a-priori boundedness is assumed, as usually in the literature, since we are also able to prove a general interpolation inequality, in turn giving local boundedness, which is valid even for weak subsolutions with no sign assumptions.

To our knowledge, this is the very first time that a strong Harnack inequality is proven for kinetic integro-differential-type equations.

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

Keywords: Harnack, nonlocal tail, Kolmogorov-Fokker-Planck, Boltzmann, Kinetic covering, Besicovitch, Slanted cylinders

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