The Well-posedness And Asymptotic Limit Theory For The Related Models In Plasma Physics

This paper mainly studies the well-posedness and its limit theory of some hydrodynamic related models in plasma physics.As is known to all,the Navier-Stokes equation is a classic hydrodynamic model derived from the law of physical conservation,which reflects the basic mechanics of viscous fluid flow.With the development of mathematical theoretical research,physicists have proposed more elaborate models.In the past two decades,quantum hydrodynamic equations and related models have also attracted great interest.In this paper,we will strictly prove the existence and decay rates of the global solutions of the quantum magnetohydrodynamic model from the theoretical analysis,the existence and decay rate of the global solutions of the full quantum hydrodynamic model,and asymptotic limit problem for the compressible Navier-Stokes-Poisson equations of Korteweg type in the half-space.Moreover,we consider the existence of global weak solutions for a class of spin-polarized ferromagnetic chain equations.This article is divided into the following six chapters.In Chapter 1,introduction.We introduce the physical background,the models and the relevant research.In Chapter 2,we study the quantum magnetohydrodynamic model for quantum plasmas.We combine the Fourier frequency method with the uniform energy estimates to obtain the existence and optimal decay rates for the solution under the small initial value perturbations.In the process of deriving energy estimates,since the quantum effect term in the momentum equation is a strong nonlinear term,this dispersion correction term makes it necessary for us to deal with higher-order spatial derivatives and find a suitable energy functional to close the energy inequality.The decay rates studied in this paper clearly describe the changing trend of the model's solutions.In Chapter 3,we consider the existence and optimal decay rates for the solution of the three-dimensional quantum hydrodynamic model under small initial disturbances.This process is very different from the previous chapter.Firstly,the model not only performs a quantum correction on the pressure tensor in the momentum equation but also a corresponding quantum correction on the energy density in the energy equation.Secondly,in terms of research methods,we no longer need to combine the decay rates of the linear equation,but directly use the negative Sobolev space to obtain the existence of the solution and the optimal decay results constructing the modified energy functional.The advantage of this method is that we only need to assume that the low-order norm of the initial value is relatively small,and the results obtained are more general.Due to the complexity of the model,we need to establish the working space of the solution by constructing a triple-vertical norm to obtain some a priori energy estimates.In Chapter 4,we consider the quasi-neutral limit,zero-viscosity limits and vanishing capillarity limit for the compressible Navier-Stokes-Poisson equations of Korteweg type(NSPK)in the half-space.The system is supplemented with the Newman,Navier-slip and Dirichlet boundary conditions,respectively,for density,velocity,and electric potential.Compared with the whole space,the main difficulty is the existence of the boundary layer.We determine the existence of the approximation solutions by analyzing the well-posedness of the system that is far from the boundary and near the boundary.Moreover,to measure the regularity of functions and deal with the integration on the boundary,we need to introduce a conormal Sobolev space to obtain the uniform energy estimates in the usual Sobolev space.In this process,we can see from the rigorous derivation mathematically that the density has a strong boundary layer,while the velocity layer has a weaker boundary layer,which also allows us to get low-order energy estimates.Then,we use conormal Sobolev space to obtain the uniform energy estimates of the error.However,because the conformal Sobolev operator does not commute with the normal derivative and the existence of the capillary effect,we use the exact expression of the higher-order commutator to obtain the priori estimates.Finally,combining the local solutions of the remainder equations,the solution of the NSKP model to the solution of the Euler equation is obtained.In Chapter 5,we study the Maxwell-Landau-Lifshitz equation with spin accumulation under Dirichlet-Neumann boundary conditions in two-dimensional magnetic multilayers.We mainly use the Leray-Schauder fixed point theorem to study the existence of a global weak solution of this system.The main difficulty is that the spin polarization parameter is between 0 and 1 in our system,which has important physical significance.When the spin polarization parameter is non-zero,the spin accumulation satisfies a quasilinear equation and therefore the analysis will be much more delicate.In Chapter 6,we mainly summarize the main results of this paper and introduce our future research.