一类混合型粘弹性力学方程组的Riemann问题

Abstract

Abstract:

In this paper, we study the Riemann problem for a system of nonlinear elasticity in a crystalline polymer model. The system contains a hyperbolic-elliptic situation and is hyperbolic degenerated in another region. We use the Liu-entropy condition to analysis the properties of all the basic waves, then we construct the solutions for the Riemann problem.

Key words: System of nonlinear elasticity, Mixed type, Riemann problem, Liu-entropy condition

CLC Number:

35L60

LIU Xiao-Min, WANG Zhen. The Riemann Problem for a System of Nonlinear Elasticity of |Mixed Type[J].Acta mathematica scientia,Series A, 2013, 33(5): 801-818.

Trendmd

References

[1] Smoller J A. Shock Waves and Reaction-Diffusion Equations (Second Edition). New York: Springer-Verlag, 1994

[2] Wendroff B. The Riemann problem for materials with nonconvex equations state I: Isentropic flow. J Math Anal Appl, 1972, 38: 454--466

[3] 梁伯润. 高分子物理学(第二版). 北京:中国纺织出版社, 2000

[4] 于同隐, 何曼君等. 高聚物的粘弹性. 上海:上海科学技术出版社, 1986

[5] 沃德 I M. 固体高聚物的力学性能. 北京:科学出版社, 1980

[6] DiPerna R J. Convergence of apprpximate solutions to conservation laws. Arch Rational Mech Anal, 1983, 82: 27--70

[7] Lin P X. Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics. Trans Amer Math Soc, 1992, 329: 377--413

[8] Zhu C J. Convergence of the viscosity solutions for the system of nonlinear elasticity. J Math Anal Appl, 1997, 209: 585--604

[9] Hattori H. The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion-isothermal case.
Arch Rat Mech Anal, 1986, 92: 247--263

[10] Hattori H. The entropy rate admissibility criterion and the entropy condition for a phase transition problem: the isothermal case. SIAM J Appl Math, 2000, 31: 791--820

[11] Hsiao L. Uniqueness of admissible solutions of the Riemann problem for a system of conservation laws of mixed type. J Diff Equations, 1990, 86: 197--233

[12] Shearer M. The Riemann problem for a class of conservation laws of mixed type. J Diff Equations, 1982, 46: 426--443

[13] Slemrod M. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch Rat Mech Anal, 1983, 81: 301--315

[14] Zhen Wang, Tingting Zhang. The Riemann problem for a class of nonlinear degenerate equations. J Math Anal Appl, 2012, 394: 112--120

[15] Liu T P. The Riemann problem for general 2 *2 conservation laws. Trans Amer Math Soc, 1974, 199: 89--112

[16] Liu T P. The Riemann problem for general systems of hyperbolic conservation laws. J Diff Equations, 1975, 18: 218--234

[17] Liu T P. Admissible Solutions of Hyperbolic Conservation Laws. Providence: Amer Math Soc, 1981

[18] Lax P D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia: SIAM, 1973

The Riemann Problem for a System of Nonlinear Elasticity of |Mixed Type

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 8

Metrics

Comments

Recommended 10

[1]	Chen Tingting, Qu Aifang, Wang Zhen. The Two-Dimensional Riemann Problem for Isentropic Chaplygin Gas [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1053-1061.
[2]	Li Huahui, Shao Zhiqiang. Vanishing Pressure Limit of Riemann Solutions to the Aw-Rascle Model for Generalized Chaplygin Gas [J]. Acta mathematica scientia,Series A, 2017, 37(5): 917-930.
[3]	Li Huahui, Shao Zhiqiang. Interactions of Delta Shock Waves for the Nonsymmetric Keyfitz-Kranzer System with Split Delta Functions [J]. Acta mathematica scientia,Series A, 2017, 37(4): 714-729.
[4]	Liu Yongquan, Guo Weiping. Convergence Theorems of Composite Implied Iterative Scheme for Mixed Type Asymptotically Nonexpansive Mappings [J]. Acta mathematica scientia,Series A, 2015, 35(2): 422-440.
[5]	SHAO Zhi-Qiang. The Riemann Problem with Delta Initial Data for the Aw-Rascle Traffic Model with Chaplygin Pressure [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1353-1371.
[6]	ZHANG Ting-Ting. The Stability of the Riemann Solutions for Aw-Rascle Model [J]. Acta mathematica scientia,Series A, 2013, 33(2): 230-241.
[7]	Liu Fagui; Ge Yunfei; Li Caizhong. Linear Hyperbolic Systems with Discontinuous Coefficients [J]. Acta mathematica scientia,Series A, 2007, 27(1): 184-192.
[8]	CHEN Shou-Xin. Global Discontinuous Solutions to a Class of Generalized Riemann Problems for Quasilinear Hyperbolic Systems [J]. Acta mathematica scientia,Series A, 2003, 23(4): 419-430.