The Weak Acoustic-Poisson Limit Of Vlasov-Poisson-Boltzmann Equation

Boltzmann equation is the most basic and classical model in Kinetic equations.It has abundant physical backgroud and practical applications.Therefore,the study of the mathematical theory of Boltzmann equation is always one of the most important and challenging fields of partial differential equations.Bardos,Glose,Levermore derive the acoustic limit from the DiPerna-Lions renoremalized solution of the Boltzmann equation.The acoustic limit is not only rich in theory,but also wide in application.In this paper,the Acoustic-Poisson limit of VPB equation is derived from the DiPerna-Lions renormalized solution of VPB.Based on the renormalized solution of VPB equation,we get conservation laws of mass,momentum and energy,we also get the entropy inequality,we give the definition of fluctuation and related properties.Finally we get the acoustic-poisson limit that is derived from VPB equation and give the weak acustic-poisson limit theorem.In the first section,the general Boltzmann equation and its symbolic meaning and research background are introduced.The classical Boltzmann equation and a special Boltzmann equation under special circumstances when N=3,V₀=1/?4pi|x|?are given.The definitions and quantities of the Boltzmann equation are given.Besides,we also give the equations of Acoustic-Poisson system.In the second section,we first give the definition of dimensionless form,and con-sider the dimensionless form of each quantity in VPB equation,and transform the mass density function F,that is,F=MG.Next,the formal conservation laws of VPB equation,namely mass conservation,momentum conservation and kinetic energy conservation,are derived.Entropy inequalities are given by defining relative entropy function H?G?and entropy dissipation rate function R?G?.In the fourth section,starting from the renormalization solution defined by DiPerna-Lions,we further deduce the properties that DiPerna-Lions renormalization solution should satisfy,namely,local mass conservation,global momentum conservation,global kinetic energy conservation and entropy inequality.Furthermore,the definition and properties of the transformation g?defined as perturbation?mentioned in Section 3 are given.The properties mainly include compactness and some non-linear estimates in each space.These compactness properties and inequality estimates will play a key role in the proof of Section 5.In the fifth section,we give the main results of this paper,namely Acoustic-Poisson weak limit theorem of Vlasov-Poisson-Boltzmann equation and its proof.Firstly,the VPB equation after renormalization is given through the definition of DiPerna-Lions renormalization.Then,by using the compactness and non-linear estimates in the fourth section,we prove that all of the equations can take the limit,and thus the weak law of local conservation is obtained.Form,further proving the main results of this paper.The study of Acoustic-Poisson limit of Vlasov-Poisson-Boltzmann equation has great theoretical and practical significance.Up to now,there are still many open prob-lems worthy of our study and exploration.We will put forward some problems in the summary section.We hope that the theoretical research on these aspects will be continuously improved and broken through.