Explicit solution for a class of coupled hyperbolic systems of conservation laws

References

• Courant R, Friedrichs K. Supersonic flow and shock waves. New York (NY): Wiley-Interscience; 1948.
   Google Scholar
• Weinan E, Rykov YG, Sinai YG. Generalized variational principles, global weak solutions and behavior with randon initial data for systems of conservation laws arising in adhesion particle dynamics. Comm Phys Math. 1996;177:349–380. doi: 10.1007/BF02101897
   Web of Science ®Google Scholar
• Korchinski DJ. Solution of a Riemann problem for a system of conservation laws possessing no classical weak solution [thesis]. Adelphi University; 1977.
   Google Scholar
• Lax PD. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Philadelphia (PA): SIAM; 1973.
   Google Scholar
• Li J, Zhang T, Yang S. The two-dimensional Riemann problem in gas dynamics, Longman Scientific and Technical; 1998. (Pitman Monogr. Surv. Pure Appl. Math.; 98).
   Google Scholar
• Nedeljkov M, Oberguggenberger M. Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J Math Anal Appl. 2008;344:1143–1157. doi: 10.1016/j.jmaa.2008.03.040
   Web of Science ®Google Scholar
• Tan D, Zhang T, Zheng Y. Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J Diff Equ. 1994;112:1–32. doi: 10.1006/jdeq.1994.1093
   Web of Science ®Google Scholar
• Yang H. Riemann problems for a class of coupled hyperbolic systems of conservation laws. J Diff Equ. 1999;159:447–484. doi: 10.1006/jdeq.1999.3629
   Web of Science ®Google Scholar
• Sheng W, Zhang T. The Riemann problem for transportation equations in gas dynamics. Memoirs Amer Math Soc. 1999;137(654 doi: 10.1090/memo/0654
   Web of Science ®Google Scholar
• Yang H, Sun W. The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws. Nonlinear Anal. 2007;67:3041–3049. doi: 10.1016/j.na.2006.09.057
   Web of Science ®Google Scholar
• Huang FM. Weak solution to pressureless type system. Commun Partial Diff Equ. 2015;30:283–304. doi: 10.1081/PDE-200050026
   Google Scholar
• Dafermos C. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Ind U Math J. 1977;26:1097–1119. doi: 10.1512/iumj.1977.26.26088
   Web of Science ®Google Scholar
• Danilov VG, Shelkovich VM. Dynamics of propagation and interaction of δ-shock waves in conservation law system. J Diff Equ. 2005;221:333–381. doi: 10.1016/j.jde.2004.12.011
   Google Scholar
• Panov EY, Shelkovich VM. δ'-shock waves as a new type of solutions to system of conservation laws. J Diff Equ. 2006;228:49–86. doi: 10.1016/j.jde.2006.04.004
   Web of Science ®Google Scholar
• Oberguggenber M. Multiplication of distributions and applications to partial differential equations. Harlow: Longman; 1992. (Pitman Res. Notes Math. Ser.; Vol. 259).
   Google Scholar
• Shen C, Sun M. Interactions of delta shock waves ofr the transport equations with split delta functions. J Math Anal Appl. 2009;351:747–755. doi: 10.1016/j.jmaa.2008.11.005
   Web of Science ®Google Scholar
• Shen C, Sun M. A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system. Int J Non Linear Mech. 2018;105:105–112. doi: 10.1016/j.ijnonlinmec.2018.06.008
   Web of Science ®Google Scholar
• Sheng W, You S. Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow. J Math Pures Appl. 2018;114:29–50. doi: 10.1016/j.matpur.2017.07.019
   Web of Science ®Google Scholar
• Sun W, Yang H. Cauchy problem for a class of coupled hyperbolic systems of conservation laws. J Yunnan University (Natural Sci). 2004;26(1):6–10.
   Google Scholar
• Chen S, Li J, Zhang T. The explicit solution to the Cauchy problem of one-dimensional transport equations. Sci China (Series A). 1997;27(11):997–1008.
   Google Scholar
• Li Y, Cao Y. Second order “large particle” difference method. Sci China (Series A). 1985;8:1024–1035.
   Google Scholar
• Liu T, Smoller J. On the vacuum state for isentropic gas dynamic equations. Adv appl Math. 1980;1:345–359. doi: 10.1016/0196-8858(80)90016-0
   Google Scholar