带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题
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摘要
研究了带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题,并得到其Riemann解的整体结构.Riemann解中包含激波,稀疏波,接触间断和δ-激波.与齐次非对称Keyfitz-Kranzer方程组不同的是非齐次非对称Keyfitz-Kranzer方程组的Riemann解是非自相似的.
In this paper, we considered the Riemann problem of generalized Chaplygin gas to the non-symmetric keyfitz-kranzer equations with a Coulomb-like friction term and obtained the global structure of Riemann solution. The Riemann solution includes shock waves, rarefaction waves, contact discontinuity and delta shock waves. The difference from the homogeneous non-symmetric keyfitz-kranzer equations is Riemann solution of system(1.1)-(1.2) that is non-self similar.
In this paper, we considered the Riemann problem of generalized Chaplygin gas to the non-symmetric keyfitz-kranzer equations with a Coulomb-like friction term and obtained the global structure of Riemann solution. The Riemann solution includes shock waves, rarefaction waves, contact discontinuity and delta shock waves. The difference from the homogeneous non-symmetric keyfitz-kranzer equations is Riemann solution of system(1.1)-(1.2) that is non-self similar.
引文
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[2] Tsien H S. Two dimensional subsonic flow of compressible fluids[J]. J Aeron Sci, 1939, 6:399-407.
[3] Sen A A, Scherrer R J. Generalizing the generalized chaplygin gas[J]. Physical Review D Particles and Fields, 2005, 72(6):237-244.
[4]张楠,吴亚波,张成园等.修正的Chaplygin气体暗能量及其Om几何诊断法[J].中国科技论文,2015,10(5):532-537.
[5] Bilic N, Tupper G B, Viollier R D. Dark Matter, dark energy and the chaplygin Gas[J]. Physics Letter B, 2002, 535:17-21.
[6] Kamenshchik A, Moschella U, Pasquier V. The Chaplygin gas as a model for dark energy[J]. 2004,751:840-859.
[7] Setare M R. Holographic Chaplygin gas model[J]. Phys Lett B, 2007, 648(5-6):329-332.
[8] Bento M C, Bertolami O, Sen A A. Generalized Chaplygin gas and CMBR constraints[J]. Physical Review D, 2003, 67(6):231-232.
[9] Kamenshchik A Y, Moschella U, Pasquier V. An alternative to quintessence[J]. Physics Letters B,2001, 511(2-4):265-268.
[10] Sen A A. Generalizing the generalized Chaplygin gas[J]. Physical Review D, 2005, 72(6):237-244.
[11] Sahni V. Dark matter an dark energy[J]. Lect Notes Phys, 2004, 653:141-180.
[12] Zhang X, Wu F Q, Zhang J. New generalized Chaplygin gas as a scheme for unification of dark energy and dark matter[J]. Journal of Cosmology and Astroparticle Physics, 2005, 2006(1):731-750.
[13] Keyfitz B L, Kranzer H C. A system of non-strictly hyperbolic conservation laws arising in elasticity theory[J]. Archive for Rational Mechanics and Analysis, 1980, 72(3):219-241.
[14] Lu Y G. Existence of global entropy solutions to general system of Keyfitz-Kranzer type[J]. Journal of Functional Analysis, 2013, 264(10):2457-2468.
[15] Cheng H. Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of keyfitz-kranzer type[J]. Advances in Mathematical Physics, 2013(11):656-681.
[16] Hernández R J C. Existence of weak entropy solution for a symmetric system of keyfitz-kranzer type[J]. Rev Colomb Mat, 2013, 47(1):13-28.
[17] Guo L, Yin G. Limit of Riemann solutions to the nonsymmetric system of keyfitz-kranzer type[J].The Scientific World Journal, 2014(3):287256.
[18] Liu S, Chen F, Wang Z. Existence of global L~P solutions to a symmetric system of Keyfitz-Kranzer type[J]. Applied Mathematics Letters, 2016, 52:96-101.
[19] Faccanoni G, Mangeney A. Exact solution for granular flows[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(10):1408-1433.
[20] Shen C. The Riemann problem for the Chaplygin gas equations with a source term[J]. Zeitschrift für Angewandte Mathematik und Mechanik, 2016, 96(6):681-695.
[21] Sun M. The exact Riemann solutions to the generalized Chaplygin gas equations with friction[J].Communications in Nonlinear Sicence and Numerical Simulation, 2016, 2:342-353.
[22] Danilov V G, Shelkovich V M. Dynamics of propagation and interaction ofδ-shock waves in conservation law systems[J]. Journal of Differential Equations, 2005, 211(2):333-381.
[23] Danilov V, Shelkovich V. Delta-shock wave type solution of hyperbolic systems of conservation laws[J]. Quarterly of Applied Mathematics, 2005, 63(3):401-427.
[24] Kalisch H, Mitrovic D. Singular solutions for the shallow-water equations[J]. Ima Journal of Applied Mathematics, 2012, 77(3):340-350.
[25] Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws[J].Proceedings of the Edinburgh Mathematical Society, 2011, 55(3):711-729.
[2] Tsien H S. Two dimensional subsonic flow of compressible fluids[J]. J Aeron Sci, 1939, 6:399-407.
[3] Sen A A, Scherrer R J. Generalizing the generalized chaplygin gas[J]. Physical Review D Particles and Fields, 2005, 72(6):237-244.
[4]张楠,吴亚波,张成园等.修正的Chaplygin气体暗能量及其Om几何诊断法[J].中国科技论文,2015,10(5):532-537.
[5] Bilic N, Tupper G B, Viollier R D. Dark Matter, dark energy and the chaplygin Gas[J]. Physics Letter B, 2002, 535:17-21.
[6] Kamenshchik A, Moschella U, Pasquier V. The Chaplygin gas as a model for dark energy[J]. 2004,751:840-859.
[7] Setare M R. Holographic Chaplygin gas model[J]. Phys Lett B, 2007, 648(5-6):329-332.
[8] Bento M C, Bertolami O, Sen A A. Generalized Chaplygin gas and CMBR constraints[J]. Physical Review D, 2003, 67(6):231-232.
[9] Kamenshchik A Y, Moschella U, Pasquier V. An alternative to quintessence[J]. Physics Letters B,2001, 511(2-4):265-268.
[10] Sen A A. Generalizing the generalized Chaplygin gas[J]. Physical Review D, 2005, 72(6):237-244.
[11] Sahni V. Dark matter an dark energy[J]. Lect Notes Phys, 2004, 653:141-180.
[12] Zhang X, Wu F Q, Zhang J. New generalized Chaplygin gas as a scheme for unification of dark energy and dark matter[J]. Journal of Cosmology and Astroparticle Physics, 2005, 2006(1):731-750.
[13] Keyfitz B L, Kranzer H C. A system of non-strictly hyperbolic conservation laws arising in elasticity theory[J]. Archive for Rational Mechanics and Analysis, 1980, 72(3):219-241.
[14] Lu Y G. Existence of global entropy solutions to general system of Keyfitz-Kranzer type[J]. Journal of Functional Analysis, 2013, 264(10):2457-2468.
[15] Cheng H. Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of keyfitz-kranzer type[J]. Advances in Mathematical Physics, 2013(11):656-681.
[16] Hernández R J C. Existence of weak entropy solution for a symmetric system of keyfitz-kranzer type[J]. Rev Colomb Mat, 2013, 47(1):13-28.
[17] Guo L, Yin G. Limit of Riemann solutions to the nonsymmetric system of keyfitz-kranzer type[J].The Scientific World Journal, 2014(3):287256.
[18] Liu S, Chen F, Wang Z. Existence of global L~P solutions to a symmetric system of Keyfitz-Kranzer type[J]. Applied Mathematics Letters, 2016, 52:96-101.
[19] Faccanoni G, Mangeney A. Exact solution for granular flows[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(10):1408-1433.
[20] Shen C. The Riemann problem for the Chaplygin gas equations with a source term[J]. Zeitschrift für Angewandte Mathematik und Mechanik, 2016, 96(6):681-695.
[21] Sun M. The exact Riemann solutions to the generalized Chaplygin gas equations with friction[J].Communications in Nonlinear Sicence and Numerical Simulation, 2016, 2:342-353.
[22] Danilov V G, Shelkovich V M. Dynamics of propagation and interaction ofδ-shock waves in conservation law systems[J]. Journal of Differential Equations, 2005, 211(2):333-381.
[23] Danilov V, Shelkovich V. Delta-shock wave type solution of hyperbolic systems of conservation laws[J]. Quarterly of Applied Mathematics, 2005, 63(3):401-427.
[24] Kalisch H, Mitrovic D. Singular solutions for the shallow-water equations[J]. Ima Journal of Applied Mathematics, 2012, 77(3):340-350.
[25] Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws[J].Proceedings of the Edinburgh Mathematical Society, 2011, 55(3):711-729.