A Mathematical Study Of Some Partial Differential Equations From Theoretical Physics

In this paper,we will use the asymptotic expansion method in the singular perturbation theory,the classical energy method and variational method(includ-ing the direct variational method and the constrained variational method),the method of upper and lower solution(also known as a monotone method),moun-tain pass lemma(also known as a min-max method),and the weighted Sobolev space technique,several kinds of equations from the field of modern physics are studied in depth.Specifically,the asymptotic limit problem of the mixed lay-er equations are studied by using the asymptotic expansion and energy method from the transport equation which arising from the radiation hydrodynamics.For the BPS vortex equation from the brane-antibrane effective theory in string theo-ry,comprehensive utilization of variational method,sup-sub solution method and weighted Sobolev method,the existence of the non topological solutions and the asymptotic behavior at infinity are studied on the whole plane.For the Yang-Mills-Chern-Simons model(YMCS model)from the gauge field theory,we use the variational method and analytical technique to study the existence of the radial symmetric solution and give the asymptotic estimation at the boundary point.Finally,for the nonlinear Schr¨odinger equation derived from modern optics,we study the existence of the vortex solution(including the saddle point solution)for the steady state equation by using the constrained variational method and mountain pass lemma.This paper is divided into five chapters.In chapter 1,the physical background of the above four kinds of equations,the present situation of the study and the main results obtained in this paper are briefly introduced.For convenience,we also enumerate some of the knowledge used in this paper.In Chapter 2,we consider the diffusion limit of the small mean free path for the radiative transfer equations.To obtain the diffusion limit of small mean free path of the radiation transport equations,we first construct formal asymptotic expansions in the usual way,and then verify the validity of these expansions.We make the small mean free path?tending to zero.It is found that the boundary condition of the limit equation is not matched with the boundary condition of the original transport equation.It is necessary for us to add perturbation terms to the solution of the limit equation to make it more close to the solution of the original one,this perturbation terms are the boundary layer(mixed layer)functions(when the initial condition of the limit equation does not match the original equation,the mixed layer will appear).By asymptotic expansion,we find the equation of the perturbation terms and solve them.After estimating the perturbation terms,the diffusion limit of the nonlinear transport equation can be obtained in the bounded region[0,1]when the mean free path tends to zero.In Chapter 3,we establish an existence theorem and some properties of so-lutions for the BPS vortex equations,which arise in the effective theory of a brane-antibrane system in string theory.We first transform the BPS equation in-to a highly nonlinear elliptic equation with Dirac mass source term in R~2.Then we use weighted Sobolev space techniques and variational methods to get a pair of sup and sub solutions.Finally,the existence of bounded solutions is proved by monotone iterative method.In addition,we give the asymptotic estimation at infinity.In chapter 4,for the classical excitation nonlinear field described in the gauge field with magnetic charge and charge:Yang-Mills-Chern-Simons(YMCS)model,we first discuss its mathematical structure and derive the two point boundary value problem for its corresponding nonlinear ordinary differential equation.Secondly,the existence of local vortex solutions of the YMCS model is proved by direct variational method,and the analysis technique is used to study the properties of the solutions.Finally,the asymptotic estimation at the end point is established by using the principle of comparison.In chapter 5,We will study the nonlinear Schr¨odinger equation appearing in the modern geometry optics and establish the existence theorem of the vortex solution of the steady state equation.First,we prove the existence of the positive radial symmetric solution by the constraint minimization problem,then the lower bounds of the wave propagation constant are also given.Secondly,we use the minimax technique to prove the existence of the nontrivial solution(saddle point solution).