Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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: 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 $ \varepsilon$, 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 $ \varepsilon\searrow 0$. It is shown that this solution admits a rigorous asymptotic expansion in $ \varepsilon$ 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 $ \varepsilon = 0$), 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.

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