From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit
From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit
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从硬球动力学到斯托克斯-傅里叶方程：玻尔兹曼-梯度极限的分析
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
L. Saint
中科院分区：
文献类型：
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作者：
T. Bodineau;I. Gallagher;L. Saint
我们推导了二维空间中N个直径为$${\vareps}$$ε的硬球系统的极限动力学的线性声学和Stokes-Fourier方程，当$$N\rightarrow \infty $$N→∞，$${\vareps}\rightarrow 0$$ε→0，$$N{\vareps}=\alpha \rightarrow \infty $$Nε=α→∞时，使用线性化的Boltzmann方程作为中间步骤。我们的证明基于Lanford的策略（大型经典系统的时间演化，Springer，柏林，1975年），以及Bodineau等人开发的修剪程序。（Invent Math 203：493-553，2016）通过定量控制将收敛时间提高到所有动力学时间，这使我们能够达到流体动力学时间尺度。这里的主要新奇在于，统一的L^2L2先验估计与微妙的对称性论证相结合，以描述粒子之间渐近去相关的累积展开的形式提供了混沌的弱版本。为了排除多重重叠的可能性，还需要对重叠进行精确的几何分析。
We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $${\varepsilon }$$ε in two space dimensions, when $$N\rightarrow \infty $$N→∞, $${\varepsilon }\rightarrow 0$$ε→0, $$N{\varepsilon }=\alpha \rightarrow \infty $$Nε=α→∞, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform $$L^2$$L2 a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.
硬球系统的玻尔兹曼梯度极限：相关误差分析
影响因子：
3.1
作者：
M. Pulvirenti;S. Simonella
通讯作者：
S. Simonella