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

Published electronically:
August 1, 2007

MathSciNet review:
2354883

Full-text PDF

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.

**1.**Graham Andrews,*On the existence of solutions to the equation 𝑢_{𝑡𝑡}=𝑢_{𝑥𝑥𝑡}+𝜎(𝑢ₓ)ₓ*, J. Differential Equations**35**(1980), no. 2, 200–231. MR**561978**, 10.1016/0022-0396(80)90040-6**2.**G. Andrews and J. M. Ball,*Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity*, J. Differential Equations**44**(1982), no. 2, 306–341. Special issue dedicated to J. P. LaSalle. MR**657784**, 10.1016/0022-0396(82)90019-5**3.**Stuart S. Antman,*The paradoxical asymptotic status of massless springs*, SIAM J. Appl. Math.**48**(1988), no. 6, 1319–1334. MR**968832**, 10.1137/0148081**4.**Stuart S. Antman,*Nonlinear problems of elasticity*, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR**2132247****5.**Stuart S. Antman and Thomas I. Seidman,*Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity*, J. Differential Equations**124**(1996), no. 1, 132–185. MR**1368064**, 10.1006/jdeq.1996.0005**6.**Stuart S. Antman and Thomas I. Seidman,*The parabolic-hyperbolic system governing the spatial motion of nonlinearly viscoelastic rods*, Arch. Ration. Mech. Anal.**175**(2005), no. 1, 85–150. MR**2106258**, 10.1007/s00205-004-0341-6**7.**P. W. Bridgman,*The Physics of High Pressure*, G. Bell and Sons, 1931; Dover reprint, 1970.**8.**R. Courant and K. O. Friedrichs,*Supersonic Flow and Shock Waves*, Interscience Publishers, Inc., New York, N. Y., 1948. MR**0029615****9.**Constantine M. Dafermos,*The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity*, J. Differential Equations**6**(1969), 71–86. MR**0241831****10.**Donald A. French, Søren Jensen, and Thomas I. Seidman,*Finite element approximation of solutions to a class of nonlinear hyperbolic-parabolic equations*, Appl. Numer. Math.**31**(1999), no. 4, 429–450. MR**1719244**, 10.1016/S0168-9274(99)00004-5**11.**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****12.**Jack K. Hale,*Asymptotic behavior of dissipative systems*, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR**941371****13.**Alain Haraux,*Nonlinear evolution equations—global behavior of solutions*, Lecture Notes in Mathematics, vol. 841, Springer-Verlag, Berlin-New York, 1981. MR**610796****14.**Ya. I. Kanel', On a model system of equations of one-dimensional gas motion (in Russian),*Diff. Urav.***4**(1969) 721-734. English translation:*Diff. Eqs.***4**(1969), 374-380.**15.**O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Lineinye i kvazilineinye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821****16.**Solomon Lefschetz,*Differential equations: Geometric theory*, Second edition. Pure and Applied Mathematics, Vol. VI, Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963. MR**0153903****17.**J.-L. Lions,*Quelques méthodes de résolution des problèmes aux limites non linéaires*, Dunod; Gauthier-Villars, Paris, 1969 (French). MR**0259693****18.**Tai Ping Liu,*The free piston problem for gas dynamics*, J. Differential Equations**30**(1978), no. 2, 175–191. MR**513269**, 10.1016/0022-0396(78)90013-X**19.**R. C. MacCamy,*Existence uniqueness and stability of solutions of the equation 𝑢_{𝑡𝑡}=(∂/∂𝑥)(𝜎(𝑢ₓ)+𝜆(𝑢ₓ)𝑢_{𝑡𝑡})*, Indiana Univ. Math. J.**20**(1970/1971), 231–238. MR**0265790****20.**Robert H. Martin Jr.,*Nonlinear operators and differential equations in Banach spaces*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR**0492671****21.**Robert E. O’Malley Jr.,*Singular perturbation methods for ordinary differential equations*, Applied Mathematical Sciences, vol. 89, Springer-Verlag, New York, 1991. MR**1123483****22.**Robert L. Pego,*Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability*, Arch. Rational Mech. Anal.**97**(1987), no. 4, 353–394. MR**865845**, 10.1007/BF00280411**23.**Donald R. Smith,*Singular-perturbation theory*, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR**812466****24.**Michael Wiegner,*On the asymptotic behaviour of solutions of nonlinear parabolic equations*, Math. Z.**188**(1984), no. 1, 3–22. MR**767358**, 10.1007/BF01163868**25.**J. Patrick Wilber,*Absorbing balls for equations modeling nonuniform deformable bodies with heavy rigid attachments*, J. Dynam. Differential Equations**14**(2002), no. 4, 855–887. MR**1940106**, 10.1023/A:1020716727905**26.**J. Patrick Wilber,*Invariant manifolds describing the dynamics of a hyperbolic-parabolic equation from nonlinear viscoelasticity*, Dyn. Syst.**21**(2006), no. 4, 465–489. MR**2273689**, 10.1080/14689360600821828**27.**J. Patrick Wilber and Stuart S. Antman,*Global attractors for degenerate partial differential equations from nonlinear viscoelasticity*, Phys. D**150**(2001), no. 3-4, 177–206. MR**1820734**, 10.1016/S0167-2789(00)00220-7**28.**Shui Cheung Yip, Stuart S. Antman, and Michael Wiegner,*The motion of a particle on a light viscoelastic bar: asymptotic analysis of its quasilinear parabolic-hyperbolic equation*, J. Math. Pures Appl. (9)**81**(2002), no. 4, 283–309. MR**1967351**, 10.1016/S0021-7824(01)01227-2**29.**E. Zeidler,*Nonlinear Functional Analysis and it Applications, Vol. II/B, Nonlinear Monotone Operators*, Springer, 1990.**30.**Songmu Zheng,*Nonlinear parabolic equations and hyperbolic-parabolic coupled systems*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 76, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. MR**1375458**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
76N99,
35B41,
35C20,
35K55

Retrieve articles in all journals with MSC (2000): 76N99, 35B41, 35C20, 35K55

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.