The Green's function and large‐time behavior of solutions for the one‐dimensional Boltzmann equation
The Green's function and large‐time behavior of solutions for the one‐dimensional Boltzmann equation
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DOI：
10.1002/cpa.20011
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发表时间：
2004-12
影响因子：
3
通讯作者：
Tai-Ping Liu;Shih-Hsien Yu
中科院分区：
文献类型：
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作者：
Tai-Ping Liu;Shih-Hsien Yu
研究了玻尔兹曼方程的绿色函数的逐点行为。我们的结果揭示了方程的粒子和流体方面。粒子方面由奇异波表示。这些波由输运方程携带，并主导解的短时行为。我们设计了一个Picard型迭代来构造越来越规则的粒子波。类流体波揭示了Navier-Stokes方程类型的耗散行为，如通常通过Chapman-Enskog展开所看到的。这些波被构造为空间变量的傅立叶模式的频谱中的长波展开的一部分。类似流体的波代表了溶液的长期行为。作为应用，我们得到了当初始扰动必然光滑时收敛到全局麦克斯韦的大时间行为的逐点描述。在分析微观速度衰减和空间衰减的相互作用时，我们充分利用了硬球模型。© 2004 Wiley Periodicals，Inc.
We study the pointwise behavior of the Green function of the Boltzmann equation. Our results reveal the particle and fluid aspects of the equation. The particle aspect is represented by singular waves. These waves are carried by transport equations and dominate the short‐time behavior of the solution. We devise a Picard‐type iteration for the construction of the increasingly regular particle‐like waves. The fluid‐like waves reveal the dissipative behavior of the type of Navier‐Stokes equations as seen usually by the Chapman‐Enskog expansion. These waves are constructed as part of the long waves expansion in the spectrum of the Fourier mode for the space variable. The fluid‐like waves represent the long‐time behavior of the solution. As an application, we obtain the pointwise description of the large‐time behavior of the convergence to the global Maxwellian when the initial perturbation is necessarily smooth. In our analysis of the exchanges of the microscopic velocity decay and space decay, we make essential uses of the hard sphere models. © 2004 Wiley Periodicals, Inc.