论文标题
在动力学的庞加莱不平等和超越
On a kinetic Poincaré inequality and beyond
论文作者
论文摘要
在本文中，我们给出了动力学不平等的轨迹证明，该证据在动力学方程的de Giorgi-Nash-Moser理论中起着重要作用。由于J. Guerand和C. Mouhot [10]在多个方向上，目前的工作改善了结果。我们使用沿向量字段$ \ partial_t + v \ cdot \ nabla_x $和$ \ partial_ {v_i} $，$ i = 1，\ dots，d $，并且不依赖$ [\ partial_ {v_i {v_i part part part f cd + v \ v \ cd + v \ v \ cd + v \ cd， \ partial_ {x_i} $或基本解决方案。提出的方法还适用于更通用的低细胞方程。我们通过研究具有$ K $步骤的Kolmogorov方程来说明这一点。
In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10] in several directions. We use kinetic trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$ and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps.
