Study On Some Properties Of Weak Solutions Of Vlasov Type Equations And Transport Equation

This dissertation focuses on the energy conservation and uniqueness of weak solutions of partial differential equations,which discusses the energy conservation of weak solutions of several types of Vlasov equations and the non-uniqueness of weak solutions of transport equation.In Chapter 1,we elaborate the research background,development history and research status of Vlasov type equations,the research significance of conservation and uniqueness of weak solutions,the application of convex integration theory,and the research overview of uniqueness and non-uniqueness of weak solutions of transport equation.In Chapter 2,we consider the energy conservation of weak solutions of Vlasov-Maxwell system for the whole space case,boundary case,relativistic case and non-relativistic case respectively.We first prove the continuity in time of the energy,and then use the classical mollification-truncation technique to establish sufficient conditions for the energy conserva-tion of the weak solutions of the Vlasov-Maxwell system in the full space case.This result improves the work of[1,2].Especially for the relativistic case,we only need the local L~2-integrability of the macroscopic densityρ.For the boundary case,due to the low regularity in time,we have to mollify in time twice with different parameters to obtain the similar conclusion.In Chapter 3,we consider the moment propagation and energy conservation of weak so-lutions of Vlasov-Poisson system.By using the classical theory of velocity moment propaga-tion and the method in Chapter 2,we obtain the existence of weak solutions with conserved energy of non-relativistic Vlasov-Poisson system.Secondly,we consider the moment prop-agation of vlasov-Poisson systems with a point charge.Based on a new integral estimate,we remove the small condition in[3]and the constraint of propagation of moments with order less than 7.In Chapter 4,we consider the renormalization properties,conservation of entropy and energy conservation of weak solutions of Vlasov-Nordstr(?)m system.By using the basic knowledge of Sobolev spaces,the properties of mollification operators and classical com-mutator estimates,we establish sufficient conditions for the entropy conservation and renor-malization properties of weak solutions of Vlasov-Nordstr(?)m system,and extend the results in Chapter 2 to Vlasov-Nordstr(?)m system.This chapter improves the estimate approach as well as the conclusions in[2,4].In Chapter 5,we consider transport equation in periodic spaces with 3 or higher di-mensions.Based on the research results of convex integration theory,we discuss the non-uniqueness of weak solutions of the d≥3-dimensional transport equation.We improve the result in[5]by using the inverse divergence operator and static Mikado flows to construct the non-zero weak solution of the transport equation whose initial value is 0.In particu-lar,the uniqueness is not guaranteed even when the index of spatial regularity is lower than Di Perna-Lions index range(1/p+1/q≤1),which implies the critical condition on temporal regularity for the uniqueness of weak solutions of transport equation.Since the construction of static Mikado flows requires geometric properties of spaces with dimensions more than 3,the method in Chapter 5 cannot be applied to 2 dimensions.In Chapter 6,we consider transport equation in 2-dimensional periodic space.Using the time-space Mikado flow instead of the static Mikado flow and the intermittency technique,we discuss the non-uniqueness problem of the transport equation in the d≥2-dimensional case with low time regularity.It should be emphasized that the application of intermittency technique in the framework of time-space Mikado flows is not ideal.We cannot get a con-clusion similar to that in Chapter 5,and the regularity index is limited by the dimensions.Nevertheless,this chapter is the first attempt to apply the intermittency technique to trans-port equations in the framework of time-space Mikado flows and partial results are obtained for two dimensional case.In Chapter 7,we present some problems to be solved.