Non-formation of a vacuum state in a standing piston problem

When boundary data is introduced, additional terms are introduced into the weak formulation of the Navier–Stokes conservation law. We examine the example of single standing piston problem. The single piston problem corresponds to a fixed boundary problem.It is intuitively clear when a single piston filled with gas is pulled apart, even though gas becomes sparse in density, a vacuum state is never formed, because of viscosity. To study this rigorously, the Navier–Stokes equations are used to describe the gas's density and velocity, subject to the presence of viscosity. We prove that, given reasonable assumptions on the boundary data, vacuum states cannot form, if they are not present initially.