| The Navier-Stokes-Poisson system is widely used to simulate the motion of charge particles(e.g.electrons,holes).It consists of the compressible Navier-Stokes equations with electrostatic force governed by the self-consistent Poisson equation.It is a coupled system of hyperbolic-parabolic-elliptic equations with strong nonlinearity.Due to its physical importance,complexity,rich phenomena and mathematical challenges,there have been a lot of numerical simulations and theoretical studies on the Navier-Stokes-Poisson equations by many physicists and mathematicians.This thesis is concerned with an initial value problem of the three-dimensional NavierStokes-Poisson equations for compressible,viscous,heat-conductive fluids subject to large and non-flat doping profile in the whole space.Due to the ellipticy of the Poisson equation,it seems difficult to obtain the t-independent estimates of the Lp-norms for the derivatives of electrostatic potential.So,the previous methods used for the Navier-Stokes equations cannot be applied to the Navier-Stokes-Poisson equations directly.Moreover,since the doping profile is large and non-flat,the steady states are non-trivial,and the stationary density/electrostatic force are governed by a quasi-linear elliptic system.This will also cause some serious difficulties in the mathematical analysis.Based on the well-known local existence result and the global a priori estimates,we establish the global well-posedness of strong solutions to the Cauchy problem of the NavierStokes-Poisson equations under the conditions that the initial energy is suitably small and the steady density is strictly away from vacuum.Although the solution has small energy,its oscillations could be arbitrarily large and the interior vacuum states are allowed.The proof relies on the t-independent lower-order a priori estimates and the t-dependent higher-order a priori estimates,which will be achieved by using the standard L2-method and the specially mathematical structure of the steady states and the material derivative.To overcome the difficulties induced by the presence of electrostatic potential,we adopt the method of timepiecewise iterative argument. |
