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: Spectrum structure and behaviours of the Vlasov-Maxwell-Fokker-Planck system, (2026), preprint
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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.