Variational methods for the kinetic Fokker-Planck equation
Variational methods for the kinetic Fokker-Planck equation
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发表时间：
2019-02
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通讯作者：
D. Albritton;S. Armstrong;J. Mourrat;M. Novack
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作者：
D. Albritton;S. Armstrong;J. Mourrat;M. Novack
我们发展了一种泛函解析方法来研究Kramers和动力学Fokker-Planck方程，该方程与经典的$H^1$一致椭圆方程理论相似。特别地，我们确定了一个类似于$H^1$的函数空间，并发展了该空间中Dirichlet问题弱解的适定性理论。在保守力的情况下，我们将弱解识别为一致凸泛函的最小值。我们证明了新的Poincare和Hormander型泛函不等式，并将它们与基本能量估计（类似于Caccioppoli不等式）结合在一起，在迭代过程中得到了弱解的$C^\infty$正则性。我们还利用庞加莱型不等式给出了动力学Fokker-Planck方程解的指数收敛到平衡的初等证明，该方程反映了热方程的经典耗散估计。
We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions of the Dirichlet problem in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincare and Hormander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the $C^\infty$ regularity of weak solutions. We also use the Poincare-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation.