Abstract:We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional derivatives only in part of the variables. Under the sole assumption that the nonlocal tail in velocity of weak solutions is $p$-summable along the drift variables, we prove a local $L^2$-$L^\infty$ estimate for kinetic integral equations and a corresponding strong Harnack inequality. The tail condition is satisfied in standard kinetic regimes considered in the literature, for instance under the usual boundedness of the mass density in the Boltzmann equation without cut-off, and it is consistent with the recent counterexample by Kassmann and Weidner (Adv. Math. 2024). These estimates further lead to a geometric characterization of the Harnack inequality, in the spirit of the seminal work of Aronson and Serrin (Arch. Ration. Mech. Anal. 1967) for the local parabolic counterpart.
| Subjects: | Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2401.14182 [math.AP] |
| (or arXiv:2401.14182v5 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2401.14182 arXiv-issued DOI via DataCite |
From: Mirco Piccinini [view email]
[v1]
Thu, 25 Jan 2024 13:43:38 UTC (689 KB)
[v2]
Thu, 9 May 2024 09:27:38 UTC (1 KB) (withdrawn)
[v3]
Tue, 13 Aug 2024 15:01:08 UTC (308 KB)
[v4]
Mon, 24 Mar 2025 09:24:59 UTC (446 KB)
[v5]
Thu, 21 May 2026 18:43:54 UTC (39 KB)
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