The Cauchy problem for the BGK equation with an external force

AbstractIn this paper, we study the Cauchy problem for the BGK equation with an external force. Firstly, we establish an L∞ existence result for this equation, and obtain some weighted L∞ estimates. Then, by means of the regularizing effects to the initial datum, we construct the approximate solutions and obtain some uniform estimates of the approximate solutions. Finally by using compactness method and passing to the limit, we prove the existence theorems of the L1 and Lp solutions and establish the propagation properties of the Lp moments

Classical and Quantum Mechanical Models of Many-Particle Systems

The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and –to a lesser extend– numerical aspects of such equations

The Kinetic and Hydrodynamic Bohm Criterions for Plasma Sheath Formation

The purpose of this paper is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criterions which are required for the formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the Euler--Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from kinetic point of view, and then Riemann derived it from the Vlasov--Poisson system. In this paper, we prove the solvability of boundary value problems of the Vlasov--Poisson system. On the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov--Poisson and Euler--Poisson systems. To clarify the relation, we also study the hydrodynamic limit of solutions of the Vlasov--Poisson system.Comment: 24 pages, 2 figure

Global strong solutions in R3 for ionic Vlasov-Poisson systems

Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space R3, under minimal assumptions on the initial data and the confining potential