For the VPFP/NSP system, the low-frequency spectrum contains acoustic and diffusive branches, giving optimal decay (1+t)^-3/4 for the density-velocity perturbation and (1+t)^-5/4 for the electric field and relative velocity.
Zhong: Green’s Function and the Pointwise Behaviors of the Vlasov-Poisson-Fokker-Planck System, Acta Math
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Spectrum analysis and optimal time decay rates of a kinetic-fluid-Poisson system
For the VPFP/NSP system, the low-frequency spectrum contains acoustic and diffusive branches, giving optimal decay (1+t)^-3/4 for the density-velocity perturbation and (1+t)^-5/4 for the electric field and relative velocity.