The Study On Stability Of Solutions To Euler-Maxwell Equations And Related Models In Plasmas

In this dissertation, we study the Euler-Maxwell equations and related models in plasmas, which describe the transport of charged particles. We will study the global existence and asymptotic behaviors of Euler-Maxwell equations in plasmas by using the tools of Fourier analysis, classical energy methods, techniques of symmetrizer and some important inequalities, such as Poincare inequality, Cauchy-Schwarz inequality, Holder inequality, Hausdorff-Young inequality, interpolation inequality, Sobolev’s embedding Theorem.In Chapter 1, first we briefly show the history of plasmas. 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 global existence of the smooth periodic solutions to the bipolar compressible isentropic Euler-Maxwell system with the adiabatic exponent γ= 3 in a three dimensional torus With the help of energy methods, the global existence of smooth solutions is established with the given initial data small.In Chapter 3, we continue to study the global existence of the smooth peri-odic solutions to the bipolar compressible isentropic Euler-Maxwell equations with general adiabatic exponent γ> 1 in the torus T. With the help of energy method and the energy functional convex method, the global smooth solution with small amplitude is established near a constant equilibrium solution with asymptotic stability property, and refined the results in the previous Chapter.In Chapter 4, we study the initial problem to the unipolar non-isentropic compressible Euler-Maxwell system in the whole space R3. With the help of tools of Fourier analysis, techniques of symmetrizer and energy method, the Lq time decay rate for the global smooth solution is established. It is shown that the density and temperature of electron converge to the equilibrium states at the same rate (1+t)-11/4 in Lq norm. This phenomenon on the charge transport shows the essential relation between the non-isentropic Euler-Maxwell and the isentropic Euler-Maxwell systems.In Chapter 5, the bipolar non-isentropic compressible Euler-Maxwell system is investigated in R3, and the Lq time decay rate for the global smooth solution is established. It is shown that the total densities, total temperatures and magnetic field of two carriers converge to the equilibrium states at the same rate (1+t)-3/2+2q/3 in Lq norm. But, both the difference of densities and the difference of temperatures of two carriers decay at the rate (1+t)-2-q/1, and the velocity and electric field decay at the rate (1+t) -3/2+2q/1. This phenomenon on the charge transport shows the essential difference between the unipolar non-isentropic Euler-Maxwell and the bipolar isentropic Euler-Maxwell system.In Chapter 6, we consider the bipolar full compressible Navier-Stokes-Maxwell equations for plasmas. We investigated, by means of the techniques of symmetrizer and energy method, the Cauchy problem in the whole space. Under the assump-tion that the initial data are close to a steady state solution, we prove that the smooth solutions of this problem converge to a steady state as the time goes to infinity.The study on Euler-Maxwell equations and related models in plasmas has not only important theoretical significance, but also extensive application value.