Maximal attractor for the system of one-dimensional polytropic viscous ideal gas

Zheng, Songmu; Qin, Yuming

doi:10.1090/qam/1848536

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.

Gui Qiang Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal. 23 (1992), no. 3, 609–634. MR 1158824, DOI https://doi.org/10.1137/0523031
Jean-Michel Ghidaglia, Finite-dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 4, 365–405 (English, with French summary). MR 963105
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371
Jack K. Hale and Anízio Perissinotto Jr., Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynam. Systems Appl. 2 (1993), no. 1, 1–9. MR 1205432
Song Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z. 216 (1994), no. 2, 317–336. MR 1278427, DOI https://doi.org/10.1007/BF02572324
Song Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl. (4) 175 (1998), 253–275. MR 1748226, DOI https://doi.org/10.1007/BF01783686
Bernhard Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations 58 (1985), no. 1, 76–103. MR 791841, DOI https://doi.org/10.1016/0022-0396%2885%2990023-3
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, Prikl. Mat. Meh. 41 (1977), no. 2, 282–291 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 2, 273–282. MR 0468593, DOI https://doi.org/10.1016/0021-8928%2877%2990011-9
Takeyuki Nagasawa, On the one-dimensional motion of the polytropic ideal gas nonfixed on the boundary, J. Differential Equations 65 (1986), no. 1, 49–67. MR 859472, DOI https://doi.org/10.1016/0022-0396%2886%2990041-0
Mari Okada and Shuichi Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ. 23 (1983), no. 1, 55–71. MR 692729, DOI https://doi.org/10.1215/kjm/1250521610

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

Reinhard Racke and Songmu Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Differential Equations 134 (1997), no. 1, 46–67. MR 1429091, DOI https://doi.org/10.1006/jdeq.1996.3216
Weixi Shen and Songmu Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal. 49 (2002), no. 1, Ser. A: Theory Methods, 21–34. MR 1887910, DOI https://doi.org/10.1016/S0362-546X%2800%2900246-7
Weixi Shen, Songmu Zheng, and Peicheng Zhu, Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions, Quart. Appl. Math. 57 (1999), no. 1, 93–116. MR 1672183, DOI https://doi.org/10.1090/qam/1672183
Jürgen Sprekels and Songmu Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Phys. D 121 (1998), no. 3-4, 252–262. MR 1645431, DOI https://doi.org/10.1016/S0167-2789%2898%2900167-5
Jürgen Sprekels, Songmu Zheng, and Peicheng Zhu, Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys, SIAM J. Math. Anal. 29 (1998), no. 1, 69–84. MR 1617175, DOI https://doi.org/10.1137/S0036141096297698
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967
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