Pointwise Estimates Of Solutions For Semiconductor Models And Vacuum Problem Of Kinetic Equations

In this thesis, we mainly consider the pointwise estimate of solutions for semiconductormodels and uniform stability of solutions for two kinds of kinetic equation near vacuum.In Chapter 1, we review the physical background and some related works on semicon-duct model and kinetic theory. We also introduce the problems we will study and the mainresults.In Chapter 2, we consider Cauchy problem of the isentropic and non-isentropic Navier-Stokes-Poisson equations in multi-dimensions with small initial data. Using standard energymethod, we obtain the global existence; then after a detailed analysis on Green functionusing splitting frequency method and the complex estimates of nonlinear terms, and com-bining the energy estimates, we obtain the pointwise estimates of the solutions. As a result,we obtain the optimal -decay rate. The results imply that the electric fieldimpacts both the integrability in space and the decay rate in time. On the other hand, theelectric field impedes the propagation of acoustic wave, thus we can't obtain the generalizedHuygens'principle. We investigate the global existence and optimal L2-decay rate of theisentropic Navier-Stokes-Poisson equations with magnetic field in the last section. In the lastsection, we obtained the pointwise estimate of solutions to the bipolar case and exhibits thegeneralized Huygens'principle, which is the essential difference to the unipolar case.In Chapter 3, we study Cauchy problem of the damped Euler-Poisson equations inmulti-dimensions with small initial data. After a transformation, we find that the densitysatisfies a special wave equation which has an exponential decay rate. Then using a newenergy method with exponential weight together with the special structure of the system, wegive a simpler proof of the existence than that in [53]. Moreover, based on a detailed analysison the Green function, we obtain the pointwise estimates of the solutions. In Chapter 4, the global existence of the inelastic Boltzmann equation near vacuum in[2] is extended to the Enskog case. We mainly study the existence and uniform stability ofsolutions. The Enskog collisions seem to be even more important in the inelastic case thanin the elastic one. This is because the inelastic kinetic theory is related to small, but macro-scopic particles, and collisions related to the finite particle size are relevant. Furthermore,we extend the stability result of the Boltzmann-Enskog equation in [47] to the Enskog caseby constructing some more complicated functionals.In Chapter 5, we consider the Cauchy problem of the relativistic Enskog equation nearvacuum. Using the method in [13], we weaken the assumptions on the scattering crosssectionσin [58], at the same time, we obtain the uniform L1(x,v)-stability of the solutions .