平衡附近的非线性量子Fokker-Planck方程
Nonlinear quantum Fokker-Planck equation near equilibrium
浏览论文内容
中文总结 AI 辅助
本文研究平衡附近具有自洽宏观量的非线性量子Fokker-Planck方程，证明了三维全空间柯西问题强解的全局存在唯一性，给出非负性与泡利上界的传播性及代数衰减率，揭示其与量子朗道方程的关联。
中文摘要 AI 辅助
我们研究一类具有自洽碰撞频率、体速度和温度的非线性量子Fokker-Planck方程。与扩散和摩擦系数给定的量子Fokker-Planck方程不同，该方程的宏观量是分布函数的非线性泛函。它保持质量、动量和动能，具有量子熵耗散结构，在费米子情形下可保持泡利容许范围，其碰撞算子在形式上也与量子朗道方程相关。针对三维全空间的柯西问题，我们证明了全局量子平衡附近强解的全局时间存在性与唯一性，该证明基于微扰宏-微观能量方法，结合了微观强制性、非线性速度矩估计以及宏观耗散论证。我们进一步建立了非负性和费米子泡利上界的传播性，在初始扰动满足额外负索伯列夫假设的条件下，得到了趋向平衡的代数衰减率。
英文摘要
We investigate a nonlinear quantum Fokker--Planck equation with self-consistent collision frequency, bulk velocity, and temperature. In contrast to quantum Fokker--Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function. The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case. Its collision operator is also formally connected to the quantum Landau equation. For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium. The proof is based on a perturbative macro--micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument. We further establish the propagation of nonnegativity and the fermionic Pauli upper bound. Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.
