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Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff softpotentials $-3<\gamma<0$ in the perturbative framework around global Maxwellianstill remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2,\infty}$ approach with time decay, we solve this problem for the full range ofcutoff potentials $-3<\gamma\leq 1$. The core of this approach lies in theinterplay between the velocity weighted $H_{x,v}^2$ energy estimate with timedecay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with timedecay for the Vlasov-Poisson-Boltzmann system, which leads to the uniformestimate with respect to the Knudsen number $\varepsilon\in (0,1]$ globally intime. As a result, global strong solution is constructed and incompressibleNavier-Stokes-Fourier-Poisson limit is rigorously justified for both hard andsoft potentials. Meanwhile, this uniform estimate with respect to$\varepsilon\in (0,1]$ also yields optimal $L^2$ time decay rate and $L^\infty$time decay rate for the Vlasov-Poisson-Boltzmann system and its incompressibleNavier-Stokes-Fourier-Poisson limit. This newly introduced weighted$H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay is flexible androbust, as it can deal with both optimal time decay problems and hydrodynamiclimit problems in a unified framework for the Boltzmann equation as well as theVlasov-Poisson-Boltzmann system for the full range of cutoff potentials. It isalso expected to shed some light on the more challenging hydrodynamic limit ofthe Landau equation and the Vlasov-Poisson-Landau system.

Diffusive Limit of the Vlasov-Poisson-Boltzmann System for the Full Range of Cutoff Potentials