Study On The Partial Regularity And Limit Theory Of Some Kinds Of Partial Differential Equations

In the first part of this thesis,we mainly consider the partial regularity,local existence,uniqueness,and singularity of coupled system solutions consisting of the Landau-Lifshitz equation and Maxwell equation under the effect of spin accumulation.The Landau-Lifshitz equation is also known as the ferromagnetic chain equation.As the basic equation describing the motion of magnetic materials,it was extremely important in magnetization theory.Since it was proposed in 1935 by Landau and Lifshitz,it has been widely concerned by mathematics and physics circles and becomes one of the research hotspots.The importance of the Landau-Lifshitz equation lies not only in its contribution to magnetism,but also in its close relationship with classical harmonic mappings and heat flow,Schrodinger flow and Ginzburg-Landau equations.To analyze the magnetization mechanism of ferromagnetic media,it makes the study of existence,regularity,and singularity of solutions of the Landau-Lifshitz equation become the heat points of great concern to mathematicians and physicists.Its extremely nonlinear structure brings great difficulty to this research,with the continuous development of partial differential equation theory,the research of Landau-Lifshitz equation has made outstanding achievements,among which academician Zhou Yulin and Guo Bailing academicians have done a lot of work in this area and formed a complete theoretical body.In this paper,we will discuss the partial regularity of the Landau-Lifshitz equation under the effect of Spin accumulation,the local existence,and regularity of two-dimensional solution,and partial regularity of the three-dimensional solution of the Landau-Lifshitz-Maxwell with Spin diffusion system in Chapter 2,Chapter 3,and Chapter 4 respectively.In the second part,we considered the approximate theory of the two-fluid Euler-Poisson equations and their related systems.As the model to describe the motion of the semiconductor and plasma,the qualitative theory of the solution of the Euler-Poisson equations can help us to analyze the motion law of plasma more clearly.The Euler-Poisson system is mainly derived from the two-fluid Euler-Maxwell system of plasma dynamics model,in which the interaction of compressible ion flow,electron flow,and the self-charting electromagnetic field are extremely complex.To study the well-posedness theory of its solution,mathematicians have derived many dispersive partial differential equations using different space-time transformations.For example,the famous nonlinear partial differential equation Korteweg-de Vries(KdV)equation which describes the propagation process of long waves in shallow water,the Kadomtsev-Petviashvili(KP)equations which describes the long and small amplitude surface waves with weak nonlinear dispersion in fluid mechanics and the Zakharov-Kuznetsov(ZK)equation which describes the propagation of nonlinear ionic sound waves in magnetized plasma.The relationship between these nonlinear partial differential equations and ionic dynamical systems can help us to study the properties of ionic dynamical systems.In Chapter 5,the KdV-type limit and the ZK-Type limit of the ionic dynamics system are proved strictly.There are six chapters in this paper,and the specific contents are summarized as follows:In Chapter 1,the physical background and mathematical research progress related to the research system are mainly introduced.In Chapter 2,the partial regularity of the three-dimensional coupled spin polarization transport model is studied.Firstly,we utilize the Ginzburg-Landau approximation theory to give an approximation system for the spin polarization transport model under the spin accumulation effect.By constructing the partial regular solutions of approximating systems,combining the energy inequalities,monotone inequalities,and pointwise estimation of modular,using the basic estimation formula we can give the energy improvement on the time slice,and finally combines the standard coverage lemma and parabolic version of Morrey lemma,we can prove the partial regularity of the solution as??0.In Chapter 3,we mainly study the partial regular solutions of the twodimensional Landau-Lifshitz-Maxwell with Spin diffusion system.Firstly,we proved the local existence of the solution by Leray-Shauder fixed point theorem;Then,by Ladyzhenskaja inequality we establish the prior estimate and the uniqueness of the solution;Secondly,under the condition that the initial value is smooth,there exists a solution that is almost smooth everywhere;Finally,we showed the boundedness of the set of singular points.Here,the difficulty is not only to overcome the nonlinear term of the Landau-Lifshitz equation,but also the effect of the quasilinear term of spin accumulation.Using the Ginzburg-Landau is to approximate the nonlinear term in the Landau-Lifshitz equation,and the quasilinear term needs to be combined with Gagliardi-Nirenberg interpolation inequality for complex energy estimation.Finally,the existence of smooth solutions for general initial value problems is given by using approximation functions.In Chapter 4,we mainly study the partial regular solutions of the threedimensional Landau-Lifshitz-Maxwell with Spin diffusion system.In a smooth bounded domain,the solution of the system is partially smooth except for a closed set whose Hausdorff measure is zero for any initial value.Firstly,using the Ginzburg-Landau approximation theory we can give the approximation system.Secondly,by the Galerkin method,we prove the existence of the weak solution of the approximation system.Then,combining the energy inequality we can the energy monotone inequality on time slice and complete the proof of energy attenuation.Finally,the Holder continuity proof of gradient is given by using the parabolic hole filling technique.In Chapter 5,firstly,we consider the KdV-type limit of the one-dimensional ionic dynamical system.On the time scale 0(?-3/4),we obtain the ionic dynamical system with ? by the Gardner-Morikawa transform.By using the perturbation method and formal expansion,we can get the homogeneous and inhomogeneous KdV-type equation on different ? scales.Then,we rigorously prove the consistent energy estimation of the remainder equation mathematically and find a suitable energy functional to make the energy closed.Secondly,we further consider the ZK-type limit of the three-dimensional ionic dynamical system with an external magnetic field.The difference is that we consider the transformation of three directions of space and the transformation of time,also using the perturbation method and formal expansion,then we obtain the homogeneous and inhomogeneous ZKtype equation on different ? scales.Finally,we give the energy estimation of the remainder equation and find a suitable energy functional to close the energy.In Chapter 6,we state a summary of this paper and the outlook for future research work.