The compressible Euler-Poisson limit to the Vlasov-Poisson-Boltzmann system with Fermi-Dirac statistics

Abstract

We justify the compressible Euler-Poisson limit of the quantum Vlasov-Poisson-Boltzmann equations with Fermi-Dirac statistics as the Knudsen number goes to zero. The truncated Hilbert expansion and the classical $L^2\text{-}{L}^{\infty }$ framework are used to derive the validity of the hydrodynamic limits of the quantum Vlasov-Poisson-Boltzmann equations with hard sphere collision kernel. Due to the coupling effects of a self-consistent electric field, the extra $W^{1,\infty }$ estimates for some new weighted function are necessary to close the energy.