论文标题
动力学Fokker-Planck方程$ h^k $空间中的低调和全球性低纤维化
Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces
论文作者
论文摘要
本文的目的是将Villani回忆录中$ H^1 $空间的动力学Fokker-Planck方程扩展到更高级别的Sobolev空间。就像在$ l^2 $和$ h^1 $设置中一样，相关操作员的$ H^k $缺乏顽强性。为了解决这个问题，我们将使用某些精心挑选的混合条款修改通常的$ h^k $规范，并具有适当的系数，这些系数是通过$ k $构建的。同时，类似的策略，但根据时间的不同（C.F. \ cite {herau}）（通常称为Hérau的方法）可以用来证明$ h^k $中的全球性低纤维性。我们规律性估计中的指数在短时间内是最佳的。此外，与我们最近的工作\ cite {glwz}一样，这里的一般结果可以在平均场设置中应用，以使估计值独立于维度。特别是，提出了对Curie-Weiss模型的应用。
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. \cite{Herau}), usually referred as Hérau's method, can be employed to prove global hypoellipticity in $H^k$. The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work \cite{GLWZ}, the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.
