The Research On The Behavior Of Solution To The Compressible Euler-Maxwell Equation

This thesis is concerned with isentropic uipolar and bipolar Euler-Maxwell system. Firstly, the asymptotic stablity and optimal convergence rate of stationary solutions to the uipolar Euler-Maxwell system whose background ions density is nb(x) have been obtained. Furthermore, We show that the solution to Cauchy problem on two-fluid Euler-Maxwell system can be well approximated by the linear diffusion wave in the sense of Darcy law at infinite time.In Chapter2, we are concerned with the compressible uipolar Euler-Maxwell equations with a nonconstant background density (e.g. of ions) in three dimensional space. There exist stationary solutions when the background density is a small pertur-bation of a positive constant state. We first show the asymptotic stability of solutions to the Cauchy problem near the stationary state provided that the initial perturba-tion is sufficiently small. Moreover the convergence rates are obtained by combining the LP-Lq estimates for the linearized equations with time-weighted estimate.In Chapter3, we are concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler-Maxwell system with collisions when initial data are around a constant equilibrium state. The main goal is the rigorous justifica-tion of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy’s law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler-Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate (1+t)-5/4in L2norm and also the convergence of the velocities and the electric field to the corresponding asymp-totic profiles given in the sense of the geneneralized Darcy’s law with the faster rate (1+t)-7/4in L2norm. Thus, this work can be also regarded as the mathematical proof of the Darcy’s law in the context of collisional fluid plasma.