The asymptotic problem for the springlike motion of a heavy piston in a viscous gas

Authors:
Stuart S. Antman and J. Patrick Wilber

Journal:
Quart. Appl. Math. **65** (2007), 471-498

MSC (2000):
Primary 76N99; Secondary 35B41, 35C20, 35K55

DOI:
https://doi.org/10.1090/S0033-569X-07-01076-5

Published electronically:
August 1, 2007

MathSciNet review:
2354883

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper treats the classical problem for the longitudinal motion of a piston separating two viscous gases in a closed cylinder of finite length. The motion of the gases is governed by singular initial-boundary-value problems for parabolic-hyperbolic partial differential equations depending on a small positive parameter , which characterizes the ratios of the masses of the gases to that of the piston. (The equation of state giving the pressure as a function of the specific volume need not be monotone and the viscosity may depend on the specific volume.) These equations are subject to a transmission condition, which is the equation of motion of the piston. The specific volumes of the gases are shown to have a positive lower bound at any finite time. This bound leads to the theorem asserting that (under mild smoothness restrictions) the initial-boundary-value problem has a unique classical solution defined for all time. The main emphasis of this paper is the treatment of the asymptotic behavior of solutions as . It is shown that this solution admits a rigorous asymptotic expansion in consisting of a regular expansion and an initial-layer expansion. The reduced problem, for the leading term of the regular expansion (which is obtained by setting ), is typically governed by an equation with memory, rather than by an ordinary differential equation of the sort governing the motion of a mass on a massless spring. The reduced problem nevertheless has a 2-dimensional attractor on which the dynamics is governed precisely by such an ordinary differential equation.

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Additional Information

**Stuart S. Antman**

Affiliation:
Department of Mathematics, Institute for Physical Science and Technology, and Institute for Systems Research, University of Maryland, College Park, Maryland 20742-4015

Email:
ssa@math.umd.edu

**J. Patrick Wilber**

Affiliation:
Department of Theoretical and Applied Mathematics, University of Akron, Akron, Ohio 44325

Email:
pwilber@math.uakron.edu

DOI:
https://doi.org/10.1090/S0033-569X-07-01076-5

Keywords:
1-dimensional gas dynamics,
piston,
viscous gas,
quasilinear parabolic-hyperbolic system,
asymptotics,
transmission,
attractors

Received by editor(s):
July 25, 2006

Published electronically:
August 1, 2007

Additional Notes:
The work of the first author was supported in part by NSF Grant # DMS-0204505.

The work of the second author was supported in part by NSF Grant DMS-0407361.

Article copyright:
© Copyright 2007
Brown University

The copyright for this article reverts to public domain 28 years after publication.