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

Volume 25 Issue 6

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2004 > 25(6): 643-652

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.

Citation:

PDF( 378 KB)

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

FullText(HTML)

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.

Relative Articles

[1]	WANG Yuning, ZHAO Baosheng. General Solutions of Elastodynamics for 2D Isotropic Porous Media[J]. Applied Mathematics and Mechanics, 2019, 40(9): 1025-1034. doi: 10.21656/1000-0887.390235
[2]	DAI Dexuan, WANG Shaowei. Linear Stability Analysis on Thermo-Bioconvection of Gyrotactic Microorganisms in a Horizontal Porous Layer Saturated by a Power-Law Fluid[J]. Applied Mathematics and Mechanics, 2019, 40(8): 856-865. doi: 10.21656/1000-0887.390298
[3]	M.Nawaz, T.Hayat, A.Alsaedi. Dufour and Soret Effects on MHD Flow of Viscous Fluid Between Radially Stretching Sheets in a Porous Medium[J]. Applied Mathematics and Mechanics, 2012, 33(11): 1304-1319. doi: 10.3879/j.issn.1000-0887.2012.11.006
[4]	S. M. Abdel-Gaied, M. R. Eid. Natural Convection of Non-Newtonian Power-Law Fluid Over Axisymmetric and Two-Dimensional Bodies of Arbitrary Shape in a Fluid-Saturated Porous Medium[J]. Applied Mathematics and Mechanics, 2011, 32(2): 171-179. doi: 10.3879/j.issn.1000-0887.2011.02.005
[5]	F. M. Hady, F. S. Ibrahim, S. M. Abdel-Gaied, M. R. Eid. Boundary-Layer Non-Newtonian Flow Over a Vertical Plate in a Porous Medium Saturated With a Nanofluid[J]. Applied Mathematics and Mechanics, 2011, 32(12): 1472-1480. doi: 10.3879/j.issn.1000-0887.2011.12.007
[6]	Y. M. Al Badawi, H. M. Duwairi. MHD Natural Convection With Joule and Viscous Heating Effects in Iso-Flux Porous Media-Filled Enclosures[J]. Applied Mathematics and Mechanics, 2010, 31(9): 1057-1065. doi: 10.3879/j.issn.1000-0887.2010.09.006
[7]	ZHANG A-man, NI Bao-yu, SONG Bing-yue, YAO Xiong-liang. Numerical Simulation of the Bubble Breakup Phenomena in the Narrow Flow Field[J]. Applied Mathematics and Mechanics, 2010, 31(4): 420-432. doi: 10.3879/j.issn.1000-0887.2010.04.004
[8]	J. C. Umavathi, I. C. Liu, J. Prathap Kumar, D. Shaik Meera. Unsteady Flow and Heat Transfer of Porous Media Sandwiched Between Viscous Fluids[J]. Applied Mathematics and Mechanics, 2010, 31(12): 1415-1434. doi: 10.3879/j.issn.1000-0887.2010.12.003
[9]	FANG Shao-mei, JIN Ling-yu, GUO Bo-ling. Global Existence of Solutions of the Periodic Initial Value Problems for Two-Dimensional Newton-Boussinesq Equations[J]. Applied Mathematics and Mechanics, 2010, 31(4): 379-388. doi: 10.3879/j.issn.1000-0887.2010.04.001
[10]	F. G. Shehadeh, H. M. Duwairi. MHD Natural Convection in Porous Media-Filled Enclosures[J]. Applied Mathematics and Mechanics, 2009, 30(9): 1042-1048. doi: 10.3879/j.issn.1000-0887.2009.09.005
[11]	K. Hooman, A. Ejlali, F. Hooman. Entropy Generation Analysis of Thermally Developing Forced Convection in a Fluid-Saturated Porous Medium[J]. Applied Mathematics and Mechanics, 2008, 29(2): 209-216.
[12]	K. Hooman, H. Gurgenci. Effects of Temperature-Dependent Viscosity Variation on Entropy Generation,Heat,and Fluid Flow Through a Porous-Saturated Duct of Rectangular Cross-Section[J]. Applied Mathematics and Mechanics, 2007, 28(1): 61-69.
[13]	LIU Ya-cheng, XU Run-zhang, YU Tao. Wave Equations With Several Nonlinear Source Terms[J]. Applied Mathematics and Mechanics, 2007, 28(9): 1079-1086.
[14]	LIU Fa-gui. Life-Span of Classical Solutions for One Dimensional Hydromagnetic Flow[J]. Applied Mathematics and Mechanics, 2007, 28(4): 462-470.
[15]	YANG Ling-e, GUO Bo-ling, XU Hai-xiang. Inhomogeneous Initial Boundary Value Problem for Generalized Ginzburg-Landau Equations[J]. Applied Mathematics and Mechanics, 2004, 25(4): 337-344.
[16]	LI Xi-kui, ZHANG Jun-bo, ZHANG Hong-wu. Instability and Dispersivity of Wave Propagation in Inelastic Saturated/Unsaturated Porous Media[J]. Applied Mathematics and Mechanics, 2002, 23(1): 31-46.
[17]	YAN Bo, ZHANG Ru-qing. Penalty Finite Element Method for Nonlinear Dynamic Response of Viscous Fluid-Saturated Biphasic Porous Media[J]. Applied Mathematics and Mechanics, 2000, 21(12): 1247-1254.
[18]	Yan Bo, Liu Zhanfang, Zhang Xiangwei. Finite Element Analysis of Wave Propagation in Fluid-Saturated Porous Media[J]. Applied Mathematics and Mechanics, 1999, 20(12): 1235-1244.
[19]	Tan Xiaoping, Pei Juemin, Chen Junkai. Three Dimensional Simulation of Unstable Immiscible Displacement in the Porous Medium[J]. Applied Mathematics and Mechanics, 1997, 18(1): 77-84.
[20]	Ding Hao-jiang, Chen Wei-qiu, Liu Zhong. Solutions to Equations of Vbrations of Spherical and Cylindlical Shells[J]. Applied Mathematics and Mechanics, 1995, 16(1): 1-13.