MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER--STOKES EQUATIONS OF  VISCOUS AND HEAT CONDUCTIVE FLUID

doi:10.1016/S0252-9602(10)60046-X

摘要/Abstract

摘要：

This article is concerned with the existence of maximal attractors in Hⁱ (i=1,2,4) for the compressible Navier--Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ω_n in Rⁿ (n=2,3). One of the important features is that the metric spaces H⁽¹⁾, H⁽²⁾, and H⁽⁴⁾ we work with are three incomplete
metric spaces, as can be seen from the constraints θ >0 and u>0, with θ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, ^…, δ₈verifying some conditions, a sequence of closed subspaces H_δ⁽ⁱ⁾ H⁽ⁱ⁾;(i=1, 2, 4) is found, and the existence of maximal (universal) attractors in H_δ⁽ⁱ⁾; (i=1, 2, 4) is
established.

关键词: compressible Navier--Stokes equations, polytropic viscous ideal gas, spherically symmetric solutions, absorbing set, maximal attractor

Abstract:

Key words: compressible Navier--Stokes equations, polytropic viscous ideal gas, spherically symmetric solutions, absorbing set, maximal attractor

中图分类号:

35Q72

秦玉明, 宋锦萍. MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER--STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID[J]. 数学物理学报(英文版), 2010, 30(1): 289-311.

QIN Yu-Ming, SONG Jin-Ping. MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER--STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID[J]. Acta mathematica scientia,Series B, 2010, 30(1): 289-311.

参考文献

[1] Antontsev S N, Kazhikhov A V, Monakhov V N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Studies in Mathematics and its Applications, Vol 22. Amsterdam, New York: North-Holland Publishing Co, 1990

[2] Chen G. Global solutions to the compressible Navier-Stokes equations for a reacting mixture. SIAM J Math Anal, 1992, 3: 609--634

[3] Deckelnick K. L² Decay for the compressible Navier-Stokes equations in unbounded domains. Comm Partial Differ Equ, 1993, 18: 1445--1476

[4] Feireisl E. The dynamical systems approach to the Navier-Stokes equations of compressible fluid. Preprint

[5] Feireisl E. Global attractors for the Navier-Stokes equations of three-dimensional compressible flow. C R Acad Sci Paris Sèr I, 2000, 331: 35--39

[6] Foias C, Temam R. The Connection Between the Navier-Stokes Equations, Dynamical Systems and Turbulence//Crandall M G, Rabinowitz P H, Turner R E L, eds. Directions in Partial Differential Equations.
Bostion, Ma: Academic Press,1987: 55--73

[7] Fujita-Yashima H, Benabidallah R. Unicite' de la solution de l'\'equation monodimensionnelle ou a' symètrie sphèrique d'un gaz visqueux et calorifère. Rendi del Circolo Mat di Palermo Ser II, 1993, 42: 195--218

[8] Fujita-Yashima H, Benabidallah R. Equationá symètrie sphèrique d'un gaz visqueux et calorifère avec la surface libre. Annali Mat pura ed applicata, 1995, 168: 75--117

[9] Ghidaglia J M. Finite dimensional behaviour for weakly damped driven Schrödinger equations. Ann Inst Henri Poincarè, 1988, 5: 365--405

[10] Hale J K. Asymptotic Behaviour of Dissipative Systems. Mathematical Surveys and Monographs, Number 25. Providence, Rhode Island: American Mathematical Society, 1988

[11] Hale J K, Perissinotto Jr A. Global attractor and convergence for one-dimensional semilinear thermoelasticity. Dynamic Systems and Applications, 1993, 2: 1--9

[12] Hoff D. Global well-posedness of the Cauchy problem for the Navire-Stokes equations of nonisentropic flow with discontinuous initial data. J Diff Eqs, 1992, 95: 33--74

[13] Hoff D, Ziane M. Compact attractors for the Navier--Stokes equations of one-dimensional compressible flow. C R Acad Sci Paris Ser I, 1999, 328: 239--244

[14] Hoff D, Ziane M. The global attractor and finite determining nodes for the Navier--Stokes equations of compressible flow with singular initial data. Indiana Univ Math J, 2000, 49: 843--889

[15] Hsiao L, Luo T. Large-time behaviour of solutions for the outer pressure problem of a viscous heat-onductive one-dimensional real gas. Proc Roy Soc Edinburgh Sect A, 1996, 126: 1277--1296

[16] Jiang S. On the asymptotic behaviour of the motion of a viscous, heat-conducting, one-dimensional real gas. Math Z, 1994, 216: 317--336

[17] Jiang S. Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain. Comm Math Phys, 1996, 178: 339--374

[18] Jiang S. Large-time behavior of solutions to the equations of a viscous polytropic ideal gas. Ann Mate Pura Appl, 1998, 175: 253--275

[19] Kawashima S, Nishida T. Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases. J Math Kyoto Univ, 1981, 21: 825--837

[20] Kazhikhov A V. To a theory of boundary value problems for equations of one-dimensional nonstationary motion of viscous heat-conduction gases. Din Sploshn Sredy, 1981, 50: 37--62 (in Russion)

[21] Kazhikhov A V. Cauchy problem for viscous gas equations. Siberian Math J, 1982, 23: 44--49

[22] Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas. J Appl Math Mech, 1977, 41: 273--282

[23] Lions P L. Mathematical Topics in Fluid Dynamics: Vol 2: Compressible Models. Oxford Lecutre Series in Mathematics and its Applications, Vol 2, No 10. Oxford: Oxford Science Publication, 1998

[24] Matsumura A, Nishida T. The initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Proc Japan Acad Ser A, 1979, 55: 337--342

[25] Matsumura A, Nishida T. The initial boundary value problems for the equations of motion of compressible viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67--104

[26] Matsumura A, Nishida T. Initial boundary value problems for the equations of motion of general fluids//Glowinski G, Lions J L, eds. Computing Meth in Appl Sci and Engin V. Amsterdam: NorthHolland, 1982: 389--406

[27] Matsumura A, Nishida T. Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm Math Phys, 1983, 89: 445--464

[28] Matsumura A, Nishida T. The initial value problem for the equations of motion of compressible viscous and heat-conductive gases. Proc Japan Acad Ser A, 1979, 55: 337--342

[29] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67--104

[30] Nagasawa T. On the one-dimensinal motion of the polytropic ideal gas non-fixed on the boundary. J Diff Equ, 1986, 65: 49--67

[31] Nikolaev V B. On the solvability of mixed problem for one-dimensional axisymmetrical viscous gas flow. Dinamicheskie zadachi Mekhaniki sploshnoj sredy, 63 Sibirsk. Otd Acad Nauk SSSR, Inst Gidrodinamiki, 1983 (Russian)

[32] Okada M, Kawashima S. On the equations of one-dimensional motion of compressible viscous fluids. J Math Kyoto Univ, 1983, 23: 55--71

[33] Padula M. Stability properties of regular flows of heat-conducting compressible fluids. J Math Kyoto Univ, 1992, 32: 401--442

[34] Qin Y. Global existence and asymptotic behaviour of solutions to a system of equations for a nonlinear one-dimensional viscous heat-conducting real gas. Chin Ann Math, 1999, 20A(3): 343--354 (in Chinese)

[35] Qin Y. Global existence and asymptotic behaviour of solutions to nonlinear hyperbolic-parabolic coupled systems with arbitrary initial data[D]. Fudan University, 1998

[36] Qin Y. Global existence and asymptotic behaviour for the solutions to nonlinear viscous, heat-conductive, one-dimensional real gas. Adv Math Sci Appl, 2000, 10: 119--148

[37] Qin Y. Global existence and asymptotic behaviour for a viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints. Nonlinear Analysis TMA, 2001, 44: 413--441

[38] Qin Y. Global existence and asymptotic behaviour of solution to the system in one-dimensional nonlinear thermoviscoelasticity. Quart Appl Math, 2001, 59: 113--142

[39] Qin Y. Global existence and asymptotic behaviour for a viscous, heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions. Adv Diff Eqs, 2002, 7: 129--154

[40] Qin Y. Exponential stability for the compressible Navier--Stokes equations. preprint

[41] Qin Y. Exponential stability for a nonlinear one-dimensional heat-conductive viscous real gas. J Math Anal Appl, 2002, 272: 507--535

[42] Qin Y, Ma T, Cavalcanti M M, Andrade D. Exponential stability in H⁴ for the Navier--Stokes equations of compressible and heat-conductive fluid. Comm Pure Appl Anal, 2005, 4: 635--664

[43] Qin Y, Rivera J M. Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas.
Proc Roy Soc Edingburgh A, 2002, 132: 685--709

[44] Radke R, Zheng S. Global existence and asymptotic behaviour in nonlinear thermoviscoelasticity. J Diff Eqs, 1997, 134: 46--67

[45] Sell G R. Global attractors for the three-dimensional Navier-Stokes equations. J Dynam Diff Eqs, 1996, 8: 1--33

[46] When W, Zheng S. Maximal attractor for the coupled Cahn-Hilliard Equations. Nonlinear Analysis TMA, 2002, 49: 21--34

[47] Shen W, Zheng S, Zhu P. Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic system with clamped boundary conditions. Quart Appl Math, 1999, 57: 93--116

[48] Sprekels J, Zheng S. Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys. Physica D, 1998, 121: 252--262

[49] Sprekels J, Zheng S, Zhu P. Asymptotic behaviour of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys. SIAM J Math Anal, 1998, 29: 69--84

[50] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Appl Math Sci, Vol 68. New York: Springer-Verlag, 1988

[51] Valli A, Zajaczkowski W M. Navier-Stokes Equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Comm Math Phys, 1986, 103: 259--296

[52] Zheng S. Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems. Pitman Series Monographs and Surveys in Pure and Applied Mathematics, Vol 76. London: Longman Group Limited, 1995

[53] Zheng S, Qin Y. Maximal attractor for the system of one-dimensional polytropic viscous ideal gas. Quart Appl Math, 2001, 59: 579--599

[54] Zheng S, Qin Y. Universal attractors for the Navier-Stokes equations of compressible and heat-conductive fluid in bounded annular domains in Rⁿ. Arch Rational Mech Anal, 2001, 160: 153--179

[55] Zheng S, Shen W. Global solutions to the Cauchy problem of the equations of one-dimensional thermoviscoelasticity. J Partial Differ Equ, 1989, 2: 26--38

MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER--STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER--STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 4

Metrics

本文评价

推荐阅读 10

[1]	王伟伟. ON GLOBAL BEHAVIOR OF WEAK SOLUTIONS OF COMPRESSIBLE FLOWS OF NEMATIC LIQUID CRYSTALS[J]. 数学物理学报(英文版), 2015, 35(3): 650-672.
[2]	K.T. Joseph and Manas R. Sahoo. SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM[J]. 数学物理学报(英文版), 2011, 31(6): 2107-2121.
[3]	姚磊; 汪文军. COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY, VACUUM AND GRAVITATIONAL FORCE IN THE CASE OF GENERAL PRESSURE[J]. 数学物理学报(英文版), 2008, 28(4): 801-817.
[4]	郝柏林, 林国广. A REMARK ON THE ATTRACTORS OF NON-NEWTONIAN FLUIDS[J]. 数学物理学报(英文版), 2003, 23(3): 316-.