METRICS FOR PROBABILITY-DISTRIBUTIONS AND THE TREND TO EQUILIBRIUM FOR SOLUTIONS OF THE BOLTZMANN-EQUATION
METRICS FOR PROBABILITY-DISTRIBUTIONS AND THE TREND TO EQUILIBRIUM FOR SOLUTIONS OF THE BOLTZMANN-EQUATION
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DOI：
10.1007/bf02179298
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发表时间：
1995-12-01
影响因子：
1.6
通讯作者：
WENNBERG, B
中科院分区：
文献类型：
--
作者：
GABETTA, G;TOSCANI, G;WENNBERG, B
本文讨论了麦克斯韦分子空间齐次玻尔兹曼方程在角截断和无限程力作用下解的平衡趋势。解被认为是概率分布的密度。田中泛函是概率分布空间的度量，它以前曾与玻尔兹曼方程一起使用。我们的主要结果是，如果初始分布具有2 + 1阶矩，则在他的度量中收敛到平衡点是指数时间的。在证明中，我们研究了概率分布空间的几个度量之间的关系，并通过证明傅里叶变换的解至少与初始数据的傅里叶变换一样规则，将其与玻尔兹曼方程联系起来。这也被用来证明，即使初始数据只有一个二阶矩，那么积分（\upper\ > R）f（upper，t）\upper\（2）d upper--> 0作为R -->无穷大，并且这种收敛在时间上是一致的。
This paper deals with the trend to equilibrium of solutions to the space-homogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2 + epsilon, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then integral(\upsilon\ > R) f(upsilon, t) \upsilon\(2) d upsilon --> 0 as R --> infinity, and this convergence is uniform in time.