动力学-流体-Poisson系统的谱分析与最优时间衰减率
Spectrum analysis and optimal time decay rates of a kinetic-fluid-Poisson system
AI总结：
本文针对VPFP/NSP耦合系统，通过引入合适范数开展谱分析，揭示其与单独VPFP、NSP系统的谱结构差异，证明非线性系统解的全局存在性并得到最优时间衰减率，为相关模型研究提供框架。
AI中文摘要：
本文研究Vlasov-Poisson-Fokker-Planck/Navier-Stokes-Poisson（VPFP/NSP）系统的柯西问题，该系统通过依赖相对速度的摩擦力和自洽Poisson方程将VPFP系统与可压缩NSP系统耦合。受Vlasov-Poisson-Boltzmann（VPB）系统谱分析的启发，本文引入合适的范数以捕捉Poisson方程诱导的作用力效应，并对全局平衡态附近的线性化系统进行详细谱分析。结果表明，两种耦合机制使耦合系统的谱结构与单独的VPFP、NSP系统存在本质差异：更准确地说，低频谱包含一对传播速度为√[(γ+1)/2]（γ≥1）的声学分支和两个扩散分支，从而恢复了经典可压缩流体的常规声波传播。此外，本文建立了非线性系统解的全局存在性，得到最优时间衰减率为(1+t)^(-3/4)，其中电场和相对速度的衰减速率更快，为(1+t)^(-5/4)。该分析还为研究通过摩擦力和自洽场耦合的相关动力学-流体模型提供了有用框架。
英文摘要：
In this paper, we consider the Cauchy problem for the Vlasov-Poisson-Fokker-Planck/Navier-Stokes-Poisson (VPFP/NSP) system, which couples the VPFP system with the compressible NSP system through a friction force dependent on the relative velocity and a self-consistent Poisson equation. Motivated by the spectrum analysis for the Vlasov-Poisson-Boltzmann (VPB) system, we introduce a suitable norm to capture the effect of the forcing induced by the Poisson equation and give a detailed spectrum analysis of the linearized system around a global equilibrium. Our results show that the two coupling mechanisms lead to an essentially different spectrum structure of the coupled system from those of the individual VPFP and NSP systems. More precisely, the low-frequency spectrum contains a pair of acoustic branches with the propagation speed $\sqrt{\frac{γ+1}{2}}~(γ\ge1)$ and two diffusive branches, thereby restoring the usual acoustic wave propagation of classical compressible fluids. Moreover, we establish the global existence of the solution to the nonlinear system and obtain the optimal time decay rate $(1+t)^{-\frac34}$, with a faster rate $(1+t)^{-\frac54}$ for the electric field and relative velocity. The present analysis also provides a useful framework for studying related kinetic-fluid models coupled through friction and self-consistent fields.
