| In this thesis,we mainly study the hydrodynamic limits of the Boltzmann equation with electromagnetic field.We focus on the incompressible Navier-Stokes-Fourier-Poisson limit of the scaled Vlasov-Poisson-Boltzmann system in the periodic box and in the whole space,the incompressible Navier-Stokes-Fourier-Maxwell limit of the scaled two-species Vlasov-Maxwell-Boltzmann system in the whole space and optimal time-decay rate of the two-fluid incompressible Navier-Stokes-Fourier-Poisson system.Firstly,we construct the global strong solution and justifly the incompressible NavierStokes-Fourier-Poisson limit of the initial value problem of the scaled Vlasov-PossionBoltzmann system in the periodic box and in the whole space.We introduce a new Hx,v1-Wx,v1,∞ framework,which consists of the Hx,v1 energy estimate,the weighted Wx,v1,∞-estimate as well as the Hx,v1 dissipation estimate on the macroscopic part of the remainder equation.Under this new Hx,v1-Wx,v1,∞ framework,we construct the global strong solution of the initial value problem of the Vlasov-Poisson-Boltzmann system(0-4)by Hilbert expansion,and verify the incompressible Navier-Stokes-Fourier-Poisson system as the first-order approximation hydrodynamic limit of the Vlasov-Poisson-Boltzmann system(0-4).Secondly,we construct the global classical solution and justify the incompressible Navier-Stokes-Fourier-Maxwell limit of the initial value problem of the scaled two-species Vlasov-Maxwell-Boltzmann system in the whole space.By using the nonlinear higher-order energy method,we obtain the uniform boundness of the solution to the scaled two-species Vlasov-Maxwell-Boltzmann system(0-5)with respect to ε.Then we prove that it converges to the incompressible Navier-Stokes-Fourier-Maxwell system.Finally,we investigate the optimal Lp(p≥2)time-decay rate of global solutions to the two-fluid incompressible Navier-Stokes-Fourier-Poisson system By employing the spectral analysis and high-low frequency decomposition method,we obtain the optimal Lp(p≥ 2)time decay rate of global solutions to the two-fluid incompressible Navier-Stokes-Fourier-Poisson system(0-6).Our results show that the velocity’s time decay rate of this system achieves the same as that of the incompressible Navier-Stokes equation,in other words,the coupling of the self-consistent Poisson equation does not change the velocity’s time decay rate of the incompressible Navier-Stokes equation.This phenomenon is interesting,compared to the compressible Navier-Stokes equation,whose velocity’s time decay rate decreases when it is coupled with the selfconsistent Poisson equation. |
