Spectrum analysis and optimal time decay rates of a kinetic-fluid-Poisson system

Abstract

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.