Hydrodynamic Limits Of Several Types Of Boltzmann Equation With External Force Fields

Hydrodynamic limit of the Boltzmann equation creates a bond between the microscopic model and the macroscopic model of gas motion,and has important applied physics background and theoretical research significance.The goals of this dissertation is to study hydrodynamic limits of several types of Boltzmann equations with external force fields,including incompressible Navier-Stokes-Fourier limit of the boundary value problem of the steady Boltzmann equation with known external force field,incompressible Navier-Stokes-Fourier-Poisson limit of the initial value problem of the rescaled two species Vlasov-Poisson-Boltzmann system with electric field,and incompressible Navier-Stokes-Fourier-Maxwell limit of the initial value problem of the rescaled VlasovMaxwell-Boltzmann system with electromagnetic field.Firstly,we study existence of positive solutions and incompressible Navier-StokesFourier limit of the steady Boltzmann equation with in-flow boundary condition and a known external force field (?).The proof is based on a (?) framework developed by Esposito-Guo-Kim-Marra and a refined positivity-preserving scheme in deriving positivity of solutions with inflow boundary condition and external force.The incompressible Navier-Stokes-Fourier limit with Dirichlet boundary condition is justified for in-flow boundary data as small perturbation of a global Maxwellian.Next,we study global strong solutions and incompressible Navier-Stokes-Fourier limits of the rescaled two-species Vlasov-Poisson-Boltzmann system in periodic region (?).Following a new (?) framework and Hilbert expansion of solutions around global Maxwellian,we construct a global strong solution and justify the two-fluid incompressible Navier-Stokes-Fourier-Poisson limit.Finally,we study global classical solutions and incompressible Navier-Stokes-FourierMaxwell limit of the rescaled Vlasov-Maxwell-Boltzmann system with electromagnetic field in R3(?).By performing perturbation expansion around global Maxwellian and showing the uniform estimates for the fluctuations with the help of the nonlinear higher-order energy method,we establish global existence of classical solutions to the Vlasov-MaxwellBoltzmann system.Moreover,we justify that it converges to the incompressible NavierStokes-Fourier-Maxwell system.