The Study On The Stability Of Electronic-magnetic Fluid Dynamical Equations And Related Models

Electronic-Magnetic fluid dynamical models are macroscopic models for semi-conductors and plasmas,which consist of Euler-Maxwell systems and Navier-Stokes-Maxwell equations.Mathematically,the study on Electronic-Magnetic fluid dynamical models is unfolded from two main aspects:the well-posedness of the model itself and the asymptotic limits between models.Based on an induc-tion of mixed time-space derivatives,techniques of anti-symmetric matrix,suitable choosing a symmetrizer and the wigner transform method,we study the stability of two fluid non-isentropic Euler-Maxwell systems with temperature diffusion,two fluid non-isentropic Euler-Poisson system and Navier-Stokes-Maxwell equations,and the combined non-relativistic and semi-classical limits for a particle defocusing the Schr?ding-Maxwell system in the whole space.In Chapter 1,first we briefly show the history of Electronic-Magnetic fluid dynamics.Then we introduce the models and their research progress.Finally the structure of this dissertation and the main research contents are presented.In Chapter 2,we mainly study the periodic problem for two fluid non-isentropic Euler-Maxwell equations with damping terms in a three dimensional torus T=?R/2??~3.By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates,the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property.Furthermore,we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for two fluid non-isentropic Euler-Poisson equations.In Chapter 3,we study the periodic problem for two-fluid non-isentropic Euler-Poisson equations in semiconductor.By choosing a suitable symmetriz-ers and using an induction argument on the order of the time-space derivatives of solutions in energy estimates,we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for two fluid non-isentropic Euler-Poisson equations.This improves the results obtained in Chapter2 for models with temperature diffusion terms.In chapter 4,we study the stability of smooth solutions near non-constant equilibrium states for a two fluid full compressible Navier-Stokes-Maxwell system in a three-dimensional torus T=(R/Z)~3.This system is quasilinear hyperbolic-parabolic.In the first part,by using the maximum principle,we find a non-constant steady-state solution with small amplitude for this system.In the second part,with the help of suitable choices of symmetrizers and classic energy estimates,we prove that global smooth solutions exist and converge to the non-constant steady-states as the time goes to infinity.In chapter 5,we prove the combined non-relativistic and semi-classical limit for a particle defocusing the Schr?ding-Maxwell system in the whole space.The electric charge and current densities,defined by the solution of Schr?ding-Maxwell equations,converge to the solution of the pressureless Euler-Poisson system.The corresponding Wigner function of the Schr?ding-Maxwell system converges to a solution of a Vlasov-Poisson system.