AI 中文总结
本文针对动力学福克 - 普朗克方程的绍德尔估计构造反例，反驳相关猜想，给出零外力变体，还在\(d\geq2\)时给出端点示例，展示有界与速度无关外力下二阶速度导数非局部有界的有界分布平稳解。
AI 中文摘要
本文针对动力学福克 - 普朗克方程的绍德尔估计构造了反例，反驳了文献[HW]中的猜想1.3。该猜想估计仅利用系数和外力的速度赫尔德正则性来控制二阶速度导数，无需空间正则性。我们还给出了零外力变体，对有界零阶系数施加相同条件。最后，在维度\(d\geq2\)时，给出端点示例表明有界与速度无关的外力可能产生二阶速度导数非局部有界的有界分布平稳解。
英文摘要
In this paper, we construct counterexamples to the Schauder estimate for kinetic Fokker--Planck equations, disproving \cite[Conjecture~1.3]{HW}. The conjectured estimate would control second velocity derivatives using only velocity Hölder regularity of the coefficients and forcing, with no spatial regularity. We also give a zero-forcing variant, where the same {conditions are imposed on} a bounded zeroth-order coefficient. Finally, in dimensions $d\geq 2$, we give endpoint examples showing that bounded velocity-independent forcing may produce bounded distributional stationary solutions whose second velocity derivatives are not locally bounded.
