J. Math. Fluid Mech. 19 (2017), 551–579
2016 Springer International Publishing
Journal of Mathematical
Analysis of a system modelling the motion of a piston in a viscous gas
Debayan Maity, Tak´eo Takahashi and Marius Tucsnak
Abstract. We study a free boundary problem modelling the motion of a piston in a viscous gas. The gas-piston system ﬁlls
a cylinder with ﬁxed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is
modeled by the 1D compressible Navier–Stokes system and the piston motion is described by the second Newton’s law.
We prove the existence and uniqueness of global in time strong solutions. The main novelty brought in by our results is
that they include the case of nonhomogeneous boundary conditions which, as far as we know, have not been studied in this
context. Moreover, even for homogeneous boundary conditions, our results require less regularity of the initial data than
those obtained in previous works.
Mathematics Subject Classiﬁcation. 35Q30, 76D05, 76N10.
Keywords. Compressible Navier–Stokes System, ﬂuid-particle interaction, strong solutions.
1. Introduction and Main Results
We consider a one dimensional model for the motion of a particle (piston) in a cylinder ﬁlled with a viscous
compressible gas. The extremities of the cylinder are ﬁxed, but the gas is allowed to penetrate inside the
cylinder. The gas is modelled by the 1D compressible Navier–Stokes equations, whereas the piston obeys
Newton’s second law. Since the position of the piston (and, consequently, the domain occupied by the gas)
is one of the unknowns of the problem, we have a free boundary value problem. Our initial motivation
was of control theoretic nature: we aimed proving that we can steer the gas to rest and the piston to any
ﬁnal position by injecting gas at the extremities of the cylinder. We intended in this way to transpose to
this physically motivated model our previous results obtained for a toy model, in which the compressible
Navier–Stokes equations are replaced by the viscous Burgers equation. We refer to V´azquez and Zuazua
[21,22] for the description and the analysis of the toy model and Liu et al. andCˆındea et al. for
the associated control problems. We quickly realized that the major diﬃculty to be solved in order to
accomplish the proposed goal consists in proving the global in time existence and uniqueness of solutions,
in appropriate function spaces. Indeed, to our knowledge, the free boundary problem we consider has not
been tackled in the literature in the case of a non vanishing gas velocity at the extremities of the cylinder
(i.e., with non homogeneous boundary conditions). The main diﬃculties are induced by the simultaneous
presence of the two ﬁxed ends where the ﬂuid can penetrate inside the cylinder (easier to describe in an
Eulerian setting) and of the free moving impermeable piston, which is easier studied in mass Lagrangian
coordinates. In this work we combine the two approaches: we use mass Lagrangian coordinates to provide
a detailed proof of the local in time existence and uniqueness and we use the Eulerian description of the
system to show that the constructed solutions are global in time.
The ﬁrst author is member of an IFCAM-project, Indo-French Center for Applied Mathematics—UMI IFCAM, Bangalore,
India, supported by DST–IISc–CNRS—and Universit´e Paul Sabatier Toulouse III. The research of the ﬁrst author was
supported by the Airbus Group Corporate Foundation Chair in Mathematics of Complex Systems established in TIFR,