Regularity And Uniqueness Of Weak Solution To Vlasov-Poisson Equation

Vlasov-Poisson equation is an important equation in kinetics，it can be used to describethe evolution of plasmas and other physical phenomena. Since this equation plays a vitalrole in the study of kinetics，it has attracted a lot of attention of both physicists andmathematicians. In this paper, we prove the regularity of t of weak solution to the3-dimensional Vlasov-Poisson equation, which slightly improves the work of E. Horst andR. Hunze [1]. Meanwhile, we give a sufcient condition on uniqueness of weak solutionon which there are a few papers.In section2, for initial value problem to3-dimensional Vlasov-Poisson equation,we first construct approximating solutions, then obtain the compactness. In section3, we get the limit solution by the advantage of compactness. In section4, we provethe uniqueness for the solutions of3-dimensional initial value problem. In order toachieve our purpose, we properly choose test functions and norms to get a Gronwalltype inequality. Besides, we make full use of the technique of approximation and theexpression of solutions constructed by the method of characteristic curve, while doingestimates.For the problem of regularity of weak solutions, our result can not cover the casewhen the base space is chosen as square integrable space. We only solve the problemin Sobolev spaces with arbitrary negative index. Hence, if we want to consider furtherproblems, finding whether weak solutions are continuous about time in square integrablespace is a good choice.The uniqueness problem for weak solutions to Vlasov-Poisson system ia one of themost difcult problems consulting this equation. Our result cannot prove the uniquenessof weak solution obtained by [1]. The uniqueness problem remains to be a challengingand hard problem. In this paper, we only make some eforts on some side of the problem.