Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case
Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case
复制标题
库仑情况下朗道方程大时间行为的估计
DOI：
10.1007/s00205-017-1078-3
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发表时间：
2015
影响因子：
2.5
通讯作者：
Lingbing He
中科院分区：
文献类型：
--
作者：
K. Carrapatoso;L. Desvillettes;Lingbing He
本文研究了具有库仑势的空间齐次朗道方程的大时间行为。首先，我们得到了一个从下界的熵耗散D（f）的加权相对Fisher信息的f关于相关的麦克斯韦分布，这导致Cercignani的猜想的一个变种由于对数Sobolev不等式。其次，我们证明了多项式和拉伸指数矩的传播最多线性增长的时间速率。作为这些估计的一个应用，我们证明了任何（H-或弱）的解决方案的朗道方程的库仑势相关的麦克斯韦平衡与一个明确的可计算的速度收敛，假设初始数据有限的质量，能量，熵和一些更高的L1矩。更准确地说，如果初始数据有一些（足够大的）多项式L1-矩，那么我们得到一个代数衰减。如果初始数据具有拉伸指数L1矩，则我们恢复拉伸指数衰减。
This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation D(f) by weighted relative Fisher information of f with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani’s conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at-most linearly growing in-time rate. As an application of these estimates, we show the convergence of any (H- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher L1-moment. More precisely, if the initial data have some (large enough) polynomial L1-moment, then we obtain an algebraic decay. If the initial data have a stretched exponential L1-moment, then we recover a stretched exponential decay.