Approximation Theory For The Euler Euler-Poisson Equations And Related Models

The approximation theory for the Euler-Poisson equations and the related models is studied in this paper.In the hydrodynamic fluid,the Euler-Poisson equations and the related models are used to describe the semiconductor device or plasma.By the theoretical study of them,we can not only enrich the model's theory of well-posedness of the solution,but also promote us to understand more deeply the essential difference and connection between the quantum plasma model and the classical plasma model.The Euler-Poisson equations and the related models are widely used in semiconductors,plasma physics and other applied sciences.The ion Euler-Poisson equations?i.e.ion acoustic model?and electron Euler-Poisson equations are derived from the low-frequency and high-frequency oscillation part of the Euler-Maxwell system respectively.The Euler-Maxwell system is used to describe the plasma dynamics,in which the compressible ion flows and electron flows interact with its own self-consistent electromagnetic field.Even only linearization is considered,there are ion acoustic waves,Langmuir waves and light waves appear.In a nonlinear case,the Euler-Maxwell model is the origin of many well-known dispersive equations,such as the Korteweg-de Vries?KdV?equation,Kadomtsev-Petviashvili?KP?equation,Zakharov equation,Zakharov-Kuznetsov?ZK?equation and nonlinear Schr?dinger?NLS?equation.By the different time-space scaling transformations and suitable expansions formally,those dispersive equations can be obtained formally from the Euler-Maxwell system.In this paper,we will prove rigorously the quantum KdV approximation?one dimensional?and quantum KP approximation?two-dimensional?of the quantum Euler-Poisson equations,and derive NLS approximation for the ion Euler-Poisson equations and quantum Euler-Poisson equations strictly.In addition,we show the global existence and uniqueness of strong and smooth large solutions to the 3D Boussinesq-MHD system without heat diffusion.This paper is divided into seven chapters.In Chapter 1,introduction.We introduce the physical background and show the models and the relevant research.In Chapter 2,we consider the long wavelength limit of the Euler-Poisson equations with quantum effect in one-dimensional.We will obtain the quantum Korteweg-de Vries equation and the inviscid Burgers equation formally under the Gardner-Morikawa?GM?transformation.Considering the time interval of orderO(?^-3/2),when nondimensional quantum parameterH?2,the quantum KdV equation will be obtained;when the quantum parameterH?28?2,the inviscid Burgers equation will be obtained.To prove this process strictly mathematically,we first make a suitable formal expansion of the unknown functions,then the approximated equation will be obtained;secondly,we will obtain the equations for the error terms by combining the original quantum Euler-Poisson equations with the approximated equation.We mainly apply the priori estimates and define energy functional to obtain the uniform energy estimate for the error.In the proving process,higher order partial derivatives need to be dealt due to the influence of quantum effect terms.In Chapter 3,the quantum KP equation can be obtained under different spatial scale transformations when considering the two-dimensional space?².In this chapter,we consider the quantum KP limit of the quantum Euler-Poisson equations in two-dimensional space?².The process is very different from the one-dimensional.Firstly,the scaling of two directions is different in the GM transformation,which leads to a different priori estimates and singular energy functional.Furthermore,the two directions need to be treated separately in the process of energy estimation due to the different singularities of the two spatial directions.Finally,this result can be generalized to the high dimensional.In Chapter 4,we consider the NLS limit of the ion Euler-Poisson equations in one dimensional in this chapter.The quasilinearity causes two basic problems in common in justifying the NLS approximation.One is loss of derivatives,which finally makes it not easy to close energy estimates.The other one is resonances due to the continuous spectrum of the linearized problem of Euler-Poisson system.Using the approximated formal expansion,the Normal-Form transformation and corrected energy functional,we can obtain a uniform energy estimation,which in turn rigorously proves that on the time scaleO(?^-2),the solution of the ion Euler-Poisson equations converges to the sine wave solution whose complex amplitude is the solution of the NLS equation.In Chapter 5,we discuss the NLS approximation of the quantum Euler-Poisson equations.We mainly use the space-time resonance method to deal with non-resonant regions in this chapter and define new energy forms to deal with the quasilinear terms.Different from Chapter 4,we divide the region into three parts.For the high frequency part which belongs to the non-resonance region,we apply the space-time resonance method,but not the Normal-Form transformation which lose derivatives itself.For the low frequency part in which the resonance region will appear,we construct modified energy functional by applying the Normal-Form transformation,but not use it to eliminate the quasilinear quadratic terms directly.In Chapter 6,we show the global existence and uniqueness of strong and smooth large solutions to the 3D Boussinesq-MHD system without heat diffusion.Since the temperature satisfies a transport equation,in order to get high regularity of temperature,we need use the combination of estimates about velocity and magnetic field.Moreover,our system involves a nonlinear damping term in the momentum equations due to the Brinkman-Forcheimer-extended-Darcy law of flow in porous media.In Chapter 7,we mainly summarize the main results of this paper and introduce our future research.