Existence Of Steady State Solutions For A Class Of Compressible Euler-Poisson Equations Coupled To An Annular Magnetic Field

In this paper,we mainly study the existence of steady-state solutions of a class of compressible Euler-Poisson equations coupled to annular magnetic field,that is,the existence of stellar solutions of rotating and non rotating gaseous magnetic field.Compared with previous studies,we discuss the pure circumferential magnetic field with different forms,and establish some new analysis and techniques.Firstly,we use the variational method to transform the problem into a class of energy functional under total mass constraints.Then,we find the minimization sequence of the functional in the space.If the minimization sequence exists,the existence of the corresponding steady-state solution can be obtained.In addition,through research,we also hope to solve the following three problems: 1.What is the structure of the pure circumferential magnetic field of a compressible star;2.Whether the existence results of pure annular magnetic stars are applicable to the case of gamma less than 2;3.Whether the magnetic field strength can be arbitrarily large.In this article,we will give answers to these questions.In the first chapter,the research background of the existence and stability of rotating and non rotating gaseous magnetic stellar solutions is introduced,and then the main research work of this paper and the necessary preparatory knowledge are introduced.In Chapter 2,the energy functional structure of non rotating gaseous magnetic field stars is obtained by variational method.In Chapter 3,the existence of energy functional minimization sequence is proved through several lemmas,so it is easy to prove the existence of stellar solution of non rotating gaseous magnetic field.In Chapter 4,the energy functional structure of rotating gaseous magnetic field stars is also obtained by variational method.In Chapter 5,the existence of stellar solution of rotating gaseous magnetic field is obtained by using the proof similar to that in Chapter 3.The sixth chapter is a summary of the methods used in this paper,as well as the problems worthy of our in-depth study.