论文标题
线性动力学理论中的定量几何控制
Quantitative Geometric Control in Linear Kinetic Theory
论文作者
论文摘要
我们考虑将传输和动力学变量上的线性碰撞与空间重量的一般线性动力学方程组合，可以在域的一部分上消失。所考虑的运输运营商包括外部电势和边界条件，例如镜面，扩散和麦克斯韦条件。所考虑的碰撞操作员包括线性松弛（散射）和福克 - 普兰克算子，边界条件包括镜面，扩散和麦克斯韦条件。我们证明在几何控制条件下指数稳定（光谱差距）的定量估计值。该论点是新的，完全依赖于发散操作员的轨迹和加权功能不平等。后一种功能不平等具有独立的利益，并意味着加权的Stokes和Korn不平等。我们最终表明，当方程不稳定时，对于频谱差距的存在并不总是必要的，在这种情况下证明了较弱的控制条件。
We consider general linear kinetic equations combining transport and a linear collision on the kinetic variable with a spatial weight that can vanish on part of the domain. The considered transport operators include external potential forces and boundary conditions, e.g. specular, diffusive and Maxwell conditions. The considered collision operators include the linear relaxation (scattering) and the Fokker-Planck operators and the boundary conditions include specular, diffusive and Maxwell conditions. We prove quantitative estimates of exponential stabilisation (spectral gap) under a geometric control condition. The argument is new and relies entirely on trajectories and weighted functional inequalities on the divergence operators. The latter functional inequalities are of independent interest and imply quantitatively weighted Stokes and Korn inequalities. We finally show that uniform control conditions are not always necessary for the existence of a spectral gap when the equation is hypoelliptic, and prove weaker control conditions in this case.
