AI 中文总结
该研究从带玻色-爱因斯坦或费米-狄拉克统计的非线性量子Fokker-Planck方程，推导得到保留量子统计效应的不可压缩Navier-Stokes-Fourier极限，明确了相关模式的紧性与声学模式的色散消失特性。
AI 中文摘要
我们从具有玻色-爱因斯坦（Bose-Einstein）或费米-狄拉克（Fermi-Dirac）统计的非线性量子Fokker-Planck方程推导不可压缩Navier-Stokes-Fourier极限。该模型具有自洽碰撞结构，局部密度充当碰撞频率，体速度和温度由分布的非线性量子加权矩确定。我们在扩散标度下的全局量子平衡附近工作，并保持量子参数固定。关于克努森数（Knudsen number）的一致估计产生强微观弛豫，并确定极限无穷小量子平衡。利用局部守恒律，我们证明不可压缩条件、布辛涅斯克（Boussinesq）关系以及无散速度分量和量子适配热模式的强紧性，而声学模式通过色散估计在局部消失。通过求解线性化量子Fokker-Planck算子的辅助方程并展开局部量子平衡流形，确定极限粘性应力张量和热通量。所得不可压缩Navier-Stokes-Fourier系统通过其归一化常数和输运系数保留量子统计的影响。
英文摘要
We derive the incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation with Bose-Einstein or Fermi-Dirac statistics. The model has a self-consistent collision structure, with the local density acting as the collision frequency and the bulk velocity and temperature determined by nonlinear quantum-weighted moments of the distribution. We work near a global quantum equilibrium under the diffusive scaling and keep the quantum parameter fixed. Uniform estimates with respect to the Knudsen number yield strong microscopic relaxation and identify the limiting infinitesimal quantum equilibrium. Using the local conservation laws, we prove the incompressibility condition, the Boussinesq relation, and strong compactness of the divergence-free velocity component and a quantum-adapted thermal mode, while the acoustic modes vanish locally by a dispersive estimate. The limiting viscous stress tensor and heat flux are identified by solving auxiliary equations for the linearized quantum Fokker-Planck operator and by expanding the local quantum equilibrium manifold. The resulting incompressible Navier-Stokes-Fourier system retains the effect of quantum statistics through its normalization constants and transport coefficients.
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