Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian
Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian
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DOI：
10.1007/s00205-007-0057-5
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发表时间：
2008-04
影响因子：
2.5
通讯作者：
Hai-liang Li;A. Matsumura
中科院分区：
文献类型：
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作者：
Hai-liang Li;A. Matsumura
我们考虑的福克-普朗克-玻尔兹曼方程的初值问题，即，被视为玻尔兹曼方程与一个额外的扩散项在速度空间中描述，例如，在热浴中的二元弹性碰撞粒子的运输。证明了在绝对Maxwell附近的初始数据的强解在时间上是全局存在的，并且在大时间内以范数渐近地趋于另一个依赖于时间的自相似Maxwell。在相空间中的扩散的影响进行了研究。它在速度空间中产生扩散过程，并导致宏观流体动力学观测的加热过程，加速自相似麦克斯韦平衡的解的收敛，其时间衰减速率比玻尔兹曼方程快。这一现象也观察到齐次福克-普朗克-玻尔兹曼方程，其中的时间衰减率的自相似麦克斯韦规范被证明是快于指数。此外，Fokker-Planck-Boltzmann方程显示收敛（在适当的缩放）的Boltzmann方程的过程中的零扩散极限。
We consider the initial value problem for the Fokker–Planck–Boltzmann equation namely, viewed as the Boltzmann equation with an additional diffusion term in velocity space to describe, for instance, the transport in thermal baths of binary elastic collisional particles. The strong solution for initial data near an absolute Maxwellian is proved to exist globally in time and tends asymptotically in the-norm to another time dependent self-similar Maxwellian in large time. The effect of the diffusion in phase space is investigated. It produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid-dynamic observable, accelerating the convergence of solutions to the equilibrium of a self-similar Maxwellian at a faster time-decay rate than the Boltzmann equation. This phenomena is also observed for homogeneous Fokker–Planck–Boltzmann equations, where the time-decay rate in the-norm to the self-similar Maxwellian is proved to be faster than exponential. Moreover, the Fokker–Planck–Boltzmann equation is shown to converge (under an appropriate scaling) strongly to the Boltzmann equation in the process of the zero diffusion limit.