Hölder regularity for nondivergence kinetic jump equations
Fix $d \ge 1$, $s \in (0, 1)$ and $0 < \lambda \le \Lambda$. Let $K(t,x,v,w)$ be measurable, even in $w$, and satisfy $$\lambda |w|^{-d-2s} \le K(t,x,v,w) \le \Lambda |w|^{-d-2s}.$$ Suppose a bounded $f : [0,1] \times B_1 \times \mathbb{R}^d \to \mathbb{R}$ solves, classically on $(0,1] \times B_1 \times B_1$, $$\partial_t f + v \cdot \nabla_x f = \operatorname{PV} \int_{\mathbb{R}^d} \bigl[f(t,x,v+w) - f(t,x,v)\bigr] K(t,x,v,w) \, dw + h,$$ where $h$ is bounded there. Imbert and Silvestre proved an estimate of the form below for bounded weak solutions under a coercivity condition, averaged upper bounds on the kernel and cancellation conditions on its part antisymmetric under interchanging the velocities $v$ and $v + w$; these conditions hold when, in addition, $K(t,x,v,w) = K(t,x,v+w,-w)$.
Problem. Must there be $\alpha > 0$ and $C$, depending only on $d$, $s$, $\lambda$ and $\Lambda$, for which $$\|f\|_{C^\alpha((1/2,1) \times B_{1/2} \times B_{1/2})} \le C \bigl(\|f\|_{L^\infty([0,1] \times B_1 \times \mathbb{R}^d)} + \|h\|_{L^\infty((0,1] \times B_1 \times B_1)}\bigr)?$$ Here $C^\alpha$ uses the Euclidean distance. Neither the interchange symmetry $K(t,x,v,w) = K(t,x,v+w,-w)$ nor continuity of the coefficients is assumed.
Source: Silvestre (2023), Conjecture 8.2, a nonlocal analogue of his kinetic Krylov–Safonov conjecture.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Hölder estimates, integro-differential equations, kinetic equations
References
@incollection{silvestre2023,
author = {Silvestre, Luis},
title = {Regularity estimates and open problems in kinetic equations},
booktitle = {A$^3$N$^2$M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models},
editor = {Mengesha, Tadele and Salgado, Abner J.},
series = {The IMA Volumes in Mathematics and its Applications},
volume = {165},
publisher = {Springer},
address = {Cham},
year = {2023},
pages = {101--148},
doi = {10.1007/978-3-031-34089-5_3},
eprint = {2204.06401},
archivePrefix = {arXiv},
}
@article{imbert2020,
author = {Imbert, Cyril and Silvestre, Luis},
title = {The weak {H}arnack inequality for the {B}oltzmann equation without cut-off},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {22},
number = {2},
year = {2020},
pages = {507--592},
doi = {10.4171/JEMS/928},
eprint = {1608.07571},
archivePrefix = {arXiv},
}
@misc{aim-fractional-kinetic-krylov-safonov,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hölder regularity for nondivergence kinetic jump equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/365-fractional-kinetic-krylov-safonov.md},
note = {Entry 365 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00592, "Hölder regularity for nondivergence kinetic jump equations", clopen.solutions.