The Study Of The Well-posedness Of Solutions To Two Classes Of Classical Hydrodynamic Equations In Plasma

This dissertation is mainly focused on the well-posedness of solutions to two classes of classical hydrodynamic equations in plasma,specifically,the stability of the planar diffusive wave for the initial value problem of the three dimensional non-isentropic bipolar Euler-Poisson equations,local existence of strong solutions to the initial value problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heatconduction and vacuum,and the blow-up criterion for the initial boundary value problem of the two-dimensional fully compressible magnetohydrodynamic equations.The details are arranged as following:In Chapter 1,we mainly introduce the relevant background,current research status and research work of the Euler-Poisson equations and the magnetohydrodynamic equations,and make some necessary preliminaries.In Chapter 2,we consider a three-dimensional full bipolar classical hydrodynamic model.This model takes the form of non-isentropic bipolar Euler-Poisson equations with the electric field and the relaxation term added to the momentum equation.Based on the diffusive wave phenomena of the one dimensional full non-isentropic bipolar Euler-Poisson equations,we show the nonlinear stability of the planar diffusive wave for the initial value problem of the three dimensional non-isentropic bipolar Euler-Poisson system.Moreover,the algebraic convergence rates are also obtained.The study generalizes the result of Li[30]to multi-dimensional case.Furthemore,we believe that the same results also hold for the switch-on case in which the far fields of two particles’ velocity in the x1-direction are different.Indeed,using the gap function with exponential decay in Huang-Mei-Wang[21],we can show the similar results for the general switch-on case.In Chapter 3,we study the initial value problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heat-conduction and vacuum as far field density.In particular,the initial density can have compact support.We prove that the initial value problem admits a local strong solution provided both the initial density and the initial magnetic field decay not too slow at infinity.The study extends the results of Liang-Shi[49]from the full compressible Navier-Stokes equations to the MHD equations,and the requirement a ∈(1,2)in[49]is relaxed to a ∈(1,∞).Moreover,the study generalizes the results of Lü-Huang[51]from isentropic to non-isentropic.In Chapter 4,we investigate the blow-up criterion for the initial boundary value problem of the two-dimensional full compressible magnetohydrodynamic equations.When the magnetic field H satisfies the perfect conducting boundary condition H·n=curlH=0,we prove the blow-up criterion(?)(‖H‖L∞(0,t;Lb)+‖divu‖L1(0,t;L∞))=∞,which depending on both H and divu.The study improves on the conclusions of Fan-Li-Nakamura[62],specifically by relaxing ‖H‖L∞(O,t;L∞)to ‖H‖L∞(0,t;Lb)(b>2).In Chapter 5,we conclude the main works of this paper.