Several Types Of Incompressible Hydrodynamic Limits Of The Boltzmann Equation

This dissertation aims at studying several types of incompressible hydrodynamic limits of the Boltzmann equation.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.We focus on studying the incompressible Euler limit of the initial value problem of the rescaled Boltzmann equation,the incompressible Euler-Poisson limit of the initial value problem of the rescaled Vlasov-Poisson-Boltzmann system and the incompressible Navier-StokesFourier limit of the boundary value problem of the steady Boltzmann equation in an exterior domain in R3.Firstly,we study the incompressible Euler limit of the rescaled Boltzmann equation Giving the local smooth solution of the incompressible Euler equation we adopt the Lx,v2-Lx,v? framework proposed in[Guo,Decay and continuity of the Boltzmann equation in bounded domains,Arch.Ration.Mech.Anal.,2010]and construct the local weak solution of the initial value problem of the Boltzmann equation(0-4),which implies that its first-order approximation is the incompressible Euler equation.Next,we study the incompressible Euler-Poisson limit of the rescaled Vlasov-PoissonBoltzmann system For the incompressible Euler-Poisson system we first establish the local well-posedness and blow-up criterion by designing an appropriate linear iteration and combining the energy method.Then,based on the Hx,v1-Wx,v1,?framework,we construct a local strong solution of the Vlasov-Poisson-Boltzmann system(0-5),which proves that its hydrodynamic limit is the incompressible Euler-Poisson system.Finally,we study the incompressible Navier-Stokes-Fourier limit of the rescaled steady Boltzmann equation with a small external force and linear boundary condition in an exterior domain For the boundary value problem of the steady incompressible Navier-Stokes-Fourier system in an exterior domain ?c U·?xU+?xP=??xU+?,?x·U=0,U|(?)?=0,U?u,as |x|??,U·?x?=??x?,?|(?)?=?w,???,as |x|??,we first use the Galerkin approximation and the energy method to obtain the existence,uniqueness and regularity estimate.Then,employing the L2-L? framework and combining with the L3 and L6 estimate of macroscopic part,we obtain the uniform upper bound estimate of the solution to the linear steady Boltzmann equation.Finally,by designing a refined positivity-preserving scheme,we construct a unique positive solution to the steady Boltzmann equation(0-6)in an exterior domain,which justifies that its incompressible limit is the incompressible Navier-Stokes-Fourier system.