The Qualitative Analysis On Solutions For Spatially Homogeneous Kinetic Equations In Moderately Soft Potential

Kinetic equations,including Boltzmann equation and Landau equation,are important models in nonequilibrium statistical physics.In this paper,we mainly studies the well posedness of the solution of the Cauchy problem of the spatially homogeneous Boltzmann equation of moderately soft potential under the cut-off assumption and the analytic smoothing effect for the solution of the linear homogeneous Landau equation with moderately soft potential.This paper is mainly divided into three chapters.In the first chapter,we introduce the research background and basic knowledge of Boltzmann equation and Landau equation,including the definition of equation,and related function space,notations,etc.Then we will present the relevant research results,the literature review at home and abroad,as well as the main research content and structure arrangement of this paper.In the second chapter,we study the well posedness of the solution to the Cauchy problem of the spatially homogeneous Boltzmann equation of moderately soft potential under the cut-off assumption.Firstly,we mainly introduce some relevant knowledge and deal with it.We take Fourier transform of v on both sides of the spatially homogeneous Boltzmann equation to be studied at the same time.However,considering the singularity of the phase functionξ^-3-γat the origin under the soft potential,we truncate the potential function in the phase space accordingly.We will discuss and study the Cauchy problem of the Boltzmann equation with moderately soft potential.We need to use mathematical knowledge such as differential mean value theorem,translation transformation,spherical coordinate transformation to prove relevant theorems,as well as Banach fixed point theorem to prove the well posedness of the solution of spatially homogeneous Boltzmann equation under moderately soft potential.In the third chapter,we prove the analytical smoothness effect of the linear spatially homogeneous Landau equation in moderately soft potential.We give some basic lemmas of linear equation in Sobolev space,we prove some lemmas by using relevant mathematical knowledge such as Leibniz formula,Cauchy Schwartz inequality and weighted energy method,and then prove the analytical smoothness of spatially linear Landau equation by induction.