Measure-valued solution for moving piston in pressureless Euler flows by the method of integration on path space

We consider the problem of a fast moving piston in the rectilinear pressureless Euler flows (also known as the sticky-particle system). As a typical initial-boundary value problem with a moving boundary, which arises in various scenarios such as fluid flow pass obstacles and fluid-solid multi-scale coupling, we are concerned with the global existence of its measure-valued solution and the force exerted on the piston by sticky fluid particles that hit it. Upon adopting a method of path integral and generalizing an elegant framework established by Ryan Hynd for the Cauchy problem, extended to the case with a moving boundary, we show the global-in-time existence of a probability measure-valued solution to the piston problem by constructing a probability measure on the path space, through approximation of finite sticky particles. The velocity of the flow is a Borel measurable function, and the distribution of mass is given by time-dependent probability measures. One of the main difficulties lies in characterizing the impulsive force exerted on the piston. The idea is to utilize a partition of the physical space by a forward flow mapping to address the difficulty. As a result, we obtain the distribution of forces exerted on the piston by the sticky particles. In particular, we prove that this force distribution is a finite Radon measure.