Several Singular Limit Problems Under The Framework Of First-order Quasilinear Hyperbolic Systems

The thesis mainly studies several singular limits of small parameters existing in general first-order quasilinear hyperbolic systems.For a cer-tain limit,we are concerned with three kinds of problems,that is the lo-cal convergence,global convergence and global convergence rates.Gener-ally speaking,the local convergence rate can be directly obtained from the local convergence result.However,due to the differences in methods of proving local and global convergence,things get different when concerning the global convergence rates.That is the reason why we need to study the global convergence rate problems separately.In this thesis,we mainly study two categories of singular limits.The first one is the global convergence rate problems in zero-relaxation time limits of general first-order quasilin-ear hyperbolic systems,Euler-Maxwell systems and Euler-Poisson systems.The second one is the fundamental relationships between the bipolar Euler-Maxwell and Euler-Poisson systems and their corresponding unipolar mod-els.These can be formally achieved by studying the zero-electron-mass limit and infinity-ion-mass limit.The thesis is organized as follows.The first chapter is dedicated to the introduction.We introduce some general methods in proving convergence of small parameters.Besides,we give a relatively detailed illustration of some common small parameters and their corresponding singular limits in Euler-Maxwell and Euler-Poisson sys-tems,and introduce their research background.Moreover,some basic lem-mas and several important inequalities are also listed at the end of the chapter.In Chapter 2 and Chapter 3,we discuss the global convergence rate problems in zero-relaxation time limits in symmetrizable first-order quasi-linear hyperbolic systems as well as their specific models.In Chapter 2,we make these analysis on general first-order hyperbolic systems.It is worth mentioning that we can eliminate the constraints that the stream function technique can only be applicable in one-dimensional spaces when the limit-ing equation is isotropic.In Chapter 3,analysis are carried out for unipolar Euler-Maxwell and Euler-Poisson systems,of which their limiting equations are the classical drift-diffusion models.In Chapter 4,5 and 6,we mainly study the zero-electron-mass limit and infinity-ion-mass limit of bipolar Euler-Maxwell and Euler-Poisson sys-tem.In the last three chapters of this thesis,we use the limit8)₀)/8)₄)?0 to illustrate this simplification,here8)₀)and8)₄)represent the mass of an elec-tron and an ion,respectively.In Chapter 4 and Chapter 5,we study the local convergence of zero-electron-mass limit and infinity-ion-mass limit of the bipolar Euler-Poisson system.Their limiting equations are the unipolar Euler-Poisson system for ions and electrons,respectively.Moreover,for the bipolar Euler-Poisson equations,we also give a global convergence result of infinity-ion-mass limit for solutions in a neighbourhood of the constant equilibrium state.In Chapter 6,we study the local convergence and global convergence of infinity-ion-mass limit of the bipolar Euler-Maxwell system.Its limiting equation is the unipolar Euler-Maxwell system for electrons.