Kolmogorov-Fokker-Planck方程的弱解：正则性、存在性与唯一性
偏微分方程分析
2025-12-10 v3
摘要
我们证明了速度变量中具有局部或非局部扩散以及粗糙扩散系数或核的Kolmogorov-Fokker-Planck方程弱解的存在性、唯一性与正则性。我们的结果涵盖Cauchy问题，并在极小假设下允许广泛的源项类别。分析的核心是一套尖锐的Lions型动力学嵌入与Bouchut-Hörmander型正则性传递结果。我们将这些工具表述为齐次、尺度不变的形式，适用于大范围的正则性参数。
引用
@article{arxiv.2403.17464,
title = {Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness},
author = {Pascal Auscher and Cyril Imbert and Lukas Niebel},
journal= {arXiv preprint arXiv:2403.17464},
year = {2025}
}
备注
This version contains the same main results but has been reorganized in a new presentation of them, making their validity for all parameters beta>0 more explicit. Some proofs were adjusted to that change. Some corollaries are explicitly stated in the introduction and more explanation are given. A density lemma (section 2) is added and an inaccuracy in Section 8 eliminated. The modifications here have no impact on the follow-up on the construction of fundamental solutions that is already published in SIAM J. of Mathematical Analysis