One-dimensional Unilateral Free-piston Problem For Rigid Collisions On Boundaries

In this paper,the rigid collision of one-dimensional unilateral free piston is considered.At the initial moment,there is a freely moving piston in the semi-infinite narrow tube closed at the left end.The right side of the piston is filled with uniform zero-pressure flow gas,and there is a vacuum between the left side and the solid wall.On the premise that the collision between gas molecules and between gas molecules and piston is completely inelastic,and the piston is completely elastically reflected on the solid wall,we construct the global existence of a solution and obtain the path of the piston’s motion.This paper uses two methods respectively.Specifically,by regarding it as a fluid-solid mixture problem,we consider the solvability and properties of the measure solution.On the other hand,we employ the particularity of the problem,directly from the physical point of view,and utilize the law of conservation of momentum to analyze.We have the same trajectory of the piston from two methods,which proves the rationality of Radon measure solution from the point of view of a specific physical problem.The results show that for the piston which is at rest initially,if the velocity of the zero pressure flow is to the left,the piston will reciprocate left and right.When the time approaches infinity,the piston will be at rest at the boundary.Moreover,the time used in each reciprocating process is the same,and the gas condensed on the piston has the same mass.If the gas is moving to the right,the piston ends up moving uniformly to the right,away from the boundary.