Maximal attractor for the system of one-dimensional polytropic viscous ideal gas
Authors:
Songmu Zheng and Yuming Qin
Journal:
Quart. Appl. Math. 59 (2001), 579-599
MSC:
Primary 35B41; Secondary 35B30, 35B40, 35L65, 37L30, 76D03
DOI:
https://doi.org/10.1090/qam/1848536
MathSciNet review:
MR1848536
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Abstract: In this paper, the dynamics for the system of polytropic viscous ideal gas is investigated. One of the important features of this problem is that the metric spaces ${H^{\left ( 1 \right )}}$ and ${H^{\left ( 2 \right )}}$ that we work with are two incomplete metric spaces, as can be seen from the constraints $\theta > 0$ and $u > 0$ with $\theta$ and $u$ begin absolute temperature and specific volume, respectively. For any constants ${\beta _1}, {\beta _2}, {\beta _3}, {\beta _4}, {\beta _5}$ satisfying certain conditions, two sequences of closed subspaces $H_\beta ^{\left ( i \right )} \subset {H^{\left ( i \right )}} \left ( i = 1, 2 \right )$ are found, and the existence of two maximal (universal) attractors in $H_\beta ^{\left ( 1 \right )}$ and $H_\beta ^{\left ( 2 \right )}$ is proved.
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G. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal. 3, 609–634 (1992)
J. M. Ghidaglia, Finite dimensional behaviour for weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré 5, 365–405 (1988)
J. K. Hale, Asymptotic behaviour of dissipative systems, Mathematical Surveys and Monographs, Number 25, American Mathematical Society, Providence, Rhode Island, 1988
J. K. Hale and A. Perissinotto, Jr., Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynamic Systems and Applications 2, 1–9 (1993)
S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z. 216, 317–336 (1994)
S. Jiang, Large-time behaviour of solutions to the equations of a viscous polytropic ideal gas, Annali di Mat. Pura Appl. 175, 253–275 (1998)
B. Kawohl, Global existence of large solutions to initial boundary value problems for a viscous heat-conducting, one-dimensional real gas, J. Differential Equations 58, 76–103 (1985)
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. 41, 273–282 (1977)
T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Differential Equations 65, 49–67 (1986)
M. Okada and S. Kawashima, On the equation of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ. 23, 55–71 (1983)
Y. Qin, Global existence and asymptotic behaviour for the solution to nonlinear one-dimensional, heat-conductive, viscous real gas, Chinese Ann. Math. 20A, 343–354 (1999) (in Chinese)
Y. Qin, Global existence and asymptotic behaviour of solutions to nonlinear hyperbolic-parabolic coupled systems with arbitrary initial data, Ph.D. Thesis, Fudan University, 1998
R. Racke and S. Zheng, Global existence and asymptotic behaviour in nonlinear thermoviscoelasticity, J. Differential Equations 134, 46–67 (1997)
W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard Equations, to appear.
W. Shen, S. Zheng, and P. Zhu, Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic system with clamped boundary conditions, Quart. Appl. Math. 57, 93–116 (1999)
J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Physica D 121, 252–262 (1998)
J. Sprekels, S. Zheng, and P. Zhu, Asymptotic behaviour of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys, SIAM J. Math. Anal. 29, 69–84 (1998)
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., Vol. 68, Springer-Verlag, New York, 1988
S. Zheng, Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems, Pitman Series Monographs and Surveys in Pure and Applied Mathematics, Vol. 76, Longman Group Limited, London, 1995
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© Copyright 2001
American Mathematical Society