Introduction Of Research Advance For Classical Solutions Of The Vlasov-Poisson Equations

This paper is a review of the article, a brief introduction to recent years of the Vlasov-Poisson equation of progress. t∈R,x, v∈R3, then an individual star of unit mass with position x and velocity v∈R3. Here (?)xUdenotes the gradient of U with respect to x. To describe the galaxy as a whole we introduce its density f= f(t, x, v)≥0 on phase space R3×R3. The integral of f over any region of phase space gives the mass or number of particles (stars) which at that instant of time have phase space coordinates in that region. The spatial mass density p= p(t, x) induced by f determines the gravitational potential U according to Newton's law for gravity, subject to the usual boundary condition at spatial infinity. If the particles are allowed to move at relativistic speeds a first modification is to replace the Vlasov equation (1) by Here v should be viewed as momentum so that v/(?) is the corresponding relativistic velocity, like all other physical constants the speed of light is normalized to unity. The system (4), (2), (3)is called the relativistic Vlasov-Poisson system. and again one distinguishes the gravitational case y= 1 and the plasma physics caseγ= 1.According to the properties of the solutions, we classified this paper into two parts. In the first part, we give the classical results of the existence and uniqueness of the initial value problems.First, we give the local existence of the solution, a local existence and uniqueness result to the initial value problem was established by Kurth [37] and see [49] for more results.Next, there are some results about spherically symmetric solution and small data solu-tion, the first global existence result for the original problem was again obtained by Batt[5] for spherically symmetric data. In the plasma physics case global existence of classical so-lutions to the relativistic Vlasov-Poisson system has been shown for spherically symmetric and for axially symmetric initial data [16,18]. In the gravitational case blow-up occurs for that system [49]. A global existence result for the Vlasov-Poisson system with sufficiently small data by Bardos and Degond [3].Then, some global existence for general data. A first global existence result was proven by Batt [4] for a modified system where the spatial density p is regularized. The development for the Vlasov-Poisson system culminated in 1989 when independently and almost simulta-neously two different proofs for global existence of classical solutions for general data were given, one by Pfaffelmoser [48] and one by Lions and Perthame [52].And in the last, we give the results of global existence of the relativistic case and other topics. Granted that global existence holds for both the attractive and the repulsive case of the Vlasov-Poisson system one certainly expects a different behavior of the two cases for large times. Results in this direction were obtained in [12,34,47].In the second part, we focus on the stability of the classical solution:We consider firstly the stability result of the non-relativistic Vlasov-Poisson system, mainly about how to construct steady states, the existence of steady states and the energy-Casimir functionals. Then the existence of minimum is obtained by reducing the problem, and we know that the minimum is the steady state.Then the relativistic case.