Moment Propagation For Vlasov-poisson With Point Charges

There have been many studies about Vlasov-Poisson equation which is one of the mostbasic dynamic model about the plasma, such as the global existence of weak solution, theexistence and uniqueness of classical solution, the gradual behavior of steady state spaceand the moment propagation. When we put point charges into the Vlasov-Poisson model,the situation changes thoroughly, the extra field which is generated by point charges couldproduce considerably large velocities of the plasma and a large spatial density in finite time.We can put the plasma-charge model into the gravitational case and repulsive case. Ifcharges and the plasma have the same sign, it is repulsive case, otherwise the situation is thegravitational field. There are some results about the weak solution. In the repulsive case,the existence and uniqueness of the global solution has been solved in two dimensions andthree dimensions; In the gravitational case, the issue of global existence of the weak solutionhas been acquired in the two dimensions, but uniqueness remains unsolved. A propagationresult for any velocity moment of order<7in three dimensions has been obtained. In thispaper, I prove the propagation result for any velocity.In this paper, we discuss the moment propagation for weak solutions to the plasma-charge system. In order to get the conclusion, we have to provide two conditions. Firstly,∫initial data satisfying L∞bounds has a finite mass. Secondly, there exist m0>6satisfyingmR3×R3(|v|2+C|x ξ0|)0/2f0(x, v)dxdv <∞. In order to prove the main conclusion, we usethe pointwise energy to control growth rate along with time. The paper consists of threefollowing parts.In the first chapter, we describe the background and summarize the main contents ofthe paper; In the second chapter, some basic estimates are proved; In the third chapter, weget the moment propagation for weak solutions to the plasma-charge system by pointwiseestimation.