非散度型动力学方程的Harnack估计
Harnack estimates for nondivergence kinetic equations
查看机构详情
- School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
- Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University（深圳北理莫斯科大学计算数学与泛函分析学院）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文针对非散度型动力学方程，在系数仅为可测的条件下建立了弱Harnack不等式和Harnack不等式，并导出内部Hölder估计，发展了概率动力学版本的Krylov--Safonov方法，是该领域首个此类正则性理论。
中文摘要 AI 辅助
我们建立了非散度型动力学方程\\[ \partial_tu+v\cdot\nabla_xu+a(t,x,v):\nabla_v^2u +b(t,x,v)\cdot\nabla_vu+c(t,x,v)u=0,\\] 的非负强上解弱Harnack不等式和非负强解Harnack不等式，其中系数$a$、$b$和$c$仅为Borel可测，矩阵$a$一致椭圆，$b$和$c$有界。作为推论，我们导出了强解的内部动力学Hölder估计。我们的方法发展了Krylov--Safonov方法的概率动力学类比。主要的新要素是Krylov估计对于具有逐步可测系数的动力学Itô过程的定量版本，这进而产生了具有有界可测系数的相关动力学SDE弱解的全局存在性。这些结果似乎是仅具有可测系数的动力学方程的首个Harnack和Hölder正则性理论。
英文摘要
We establish a weak Harnack inequality for nonnegative strong supersolutions and a Harnack inequality for nonnegative strong solutions of the nondivergence-form kinetic equation \[ \partial_tu+v\cdot\nabla_xu+a(t,x,v):\nabla_v^2u +b(t,x,v)\cdot\nabla_vu+c(t,x,v)u=0, \] where the coefficients $a$, $b$, and $c$ are merely Borel measurable, the matrix $a$ is uniformly elliptic, and $b$ and $c$ are bounded. As a consequence, we derive interior kinetic Hölder estimates for strong solutions. Our approach develops a probabilistic kinetic analogue of the Krylov--Safonov method. The main new ingredient is a quantitative version of Krylov's estimate for kinetic Itô processes with progressively measurable coefficients, which in turn yields the global existence of weak solutions to the associated kinetic SDE with bounded measurable coefficients. These results appear to be the first Harnack and Hölder regularity theory for kinetic equations with merely measurable coefficients.
