Global Existence and Blow-up Phenomena of Classical Solutions for the System of Compressible Adiabatic Flow Through Porous Media

LIU Fa-gui; KONG De-xing

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LIU Fa-gui, KONG De-xing. Global Existence and Blow-up Phenomena of Classical Solutions for the System of Compressible Adiabatic Flow Through Porous Media[J]. Applied Mathematics and Mechanics, 2004, 25(6): 643-652.

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Global Existence and Blow-up Phenomena of Classical Solutions for the System of Compressible Adiabatic Flow Through Porous Media

LIU Fa-gui¹,
KONG De-xing²

1.
Department of Mathematics and Information Sciences, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou 450008, P. R. China;

Received Date: 2001-11-27
Rev Recd Date: 2003-12-08
Publish Date: 2004-06-15

Abstract

Abstract

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small' solution.
- porous media,
- compressible adiabatic flow,
- system of equations,
- classical solution,
- global existence,
- blow-up

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References(12)

References

[1]	LIU Tai-ping. Development of singularities in the nonlinear wave for quasilinear partial differential equations[J].J Differential Equations,1979,33(2):92—111. doi: 10.1016/0022-0396(79)90082-2
[2]	ZHAO Yan-chun. Global smooth solutions for one dimensional gas dynamics systems[R]. IMA Preprint #545,University of Minnesuta,1989.
[3]	KONG De-xing.Cauchy Problem for Quasilinear Hyperbolic Systems[M].MSJ Memoirs No 6.Tokyo:the Mathematical Society of Japan,2000.
[4]	KONG De-xing.Life-span of the classical solutions of nonlinear hyperbolic systems[J].J Partial Differential Equations,1996,11(2):221—240.
[5]	Nishida T.Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics[M].Paris-Sud:Publications Mathématiqées D'osay 78-02,1978.
[6]	Slemrod M. Instability of steady shearing flows in a nonlinear viscolastic fluid[J].Arch Rational Mech Anal,1978,68(3):211—225.
[7]	LIN Long-wei,ZHENG Yong-shu.Existence and nonexistence of global smooth solutions for quasilinear hyperbolic system[J].Chinese Ann Math，Ser B,1988,9(4):372—377.
[8]	王剑华，李才中.具耗散拟线性双曲型方程组整体光滑可解性与奇性形成[J].数学年刊,A辑，1988,9(5):509—523.
[9]	HSIAO Ling,Serre D.Global existence of solutions for the systems of compressible adiabatic flow through porous media[J].SIAM J Math Anal,1996,27(1):70—77. doi: 10.1137/S0036141094267078
[10]	郑永树.具耗散一维气体动力学方程组整体光滑解[J].数学年刊,A辑，1996，17(2):155—162.
[11]	KONG De-xing. Maximum principles for quasilinear hyperbolic systems and its applications[J].Nonlinear Analysis: Theory, Methods and Applications,1998,32(9):871—880. doi: 10.1016/S0362-546X(97)00534-8
[12]	LI Ta-tsien,YU Wen-ci.Boundary Value Problems for Quasilinear Hyperbolic Systems[M].Mathematics Series V,Duke University Press，1985.