Global Solutions To Vlasov-Poisson-BGK Equations With Force Fields

Most of the matter in the universe is in the plasma state,such as stars,nebulae and the surface of the sun.The plasma is a multi-particle systems which composed of charged parti-cles. Electrostatic force and collisions exist among charged particles. Vlasov-Poisson-BGK equation is an important kind of dynamics model of astrophysics and plasma physics.In this paper,we discuss the existence of global weak solutions to the Cauchy problem of the Vlasov-Poisson-BGK system with external magnetic fields under the condition that initial data belong to the space LP(Rx3×Rv3)with p>3and have finite second order veloc-ity moments,and the external magnetic field B(t,x) belong to the space Lq((O,T)×Rx3where1/p+1/q=1. We prove our main conclusions by asymptotic methods. Firstly, we discuss the existence and uniqueness of solutions to the Cauchy problem with smooth ex-ternal force fields. Secondly, we construct an approximation equation by modifying the fundamental solution of the Laplace equation. In order to solve it, we introduce a cutoff function, and modify the approximation equation by a mollifier.Then we obtain a new equa-tion,and obtain its solution by Schauder’s theorem. Because of velocity moments lemma and velocity averaging lemma, we get the existence of approximation solution by going to limit. Finally,we can obtain the compactness of approximation solution sequence by veloc-ity moments lemma and velocity averaging lemma. Then we can go to limit in the weak form of approximation equation. We get the weak solution of the Cauchy problem of the Vlasov-Poisson-BGK system with external magnetic fields.When the external magnetic field B(t,x) of the discussed Cauchy problem in this paper is zero,this problem is converted into the problem studied in [1].Therefore, the results in this paper implies Zhang’s results in [1]. The point is that the index p>9is optimized for p>3.