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The delta-shock wave for the two variables of a class of Temple system. (English) Zbl 1445.35246

The authors study the Riemann problem for the nonstrictly hyperbolic system of conservation laws \(\partial_t u+\partial_x(u^2v)=\partial_t v+\partial_x(uv^2)=0\). It is shown that each component of the solution may contain \(\delta\)-shock waves whenever \(uv=0\) for the right state of Riemann data. The authors also solve the generalized Riemann problem when the initial data contains an intermediate state supported on a small segment, and establish stability of the Riemann solution with respect to the perturbation parameter.

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35B35 Stability in context of PDEs
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[1] Ambrosio, L., Crippa, G., Figalli, A., Spinolo, L.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J. Math. Anal. 41, 1890-1920 (2009) · Zbl 1222.35060 · doi:10.1137/090754686
[2] Bianchini, S.: Stability of L∞\(L^{\infty}\) solutions for hyperbolic systems with coinciding shocks and rarefactions. SIAM J. Math. Anal. 33, 959-981 (2001) · Zbl 1009.35052 · doi:10.1137/S0036141000377900
[3] Chang, T., Hsiao, L.: The Riemann Problem and Interaction of Waves in Gas Dynamics. Pitman Monogr. Surv. Pure Appl. Math., vol. 41. Longman, Harlow (1989) · Zbl 0698.76078
[4] Chen, G.Q., Liu, H.: Formation of δ-shock and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34, 925-938 (2003) · Zbl 1038.35035 · doi:10.1137/S0036141001399350
[5] Cheng, H., Yang, H.: Delta shock waves in chromatography equations. J. Math. Anal. Appl. 380, 475-485 (2011) · Zbl 1217.35120 · doi:10.1016/j.jmaa.2011.04.002
[6] Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of δ-shock waves in conservation law systems. J. Differ. Equ. 221, 333-381 (2005) · Zbl 1072.35121 · doi:10.1016/j.jde.2004.12.011
[7] Danilov, V.G., Shelkovich, V.M.: Delta-shock type solution of hyperbolic systems of conservation law systems. Q. Appl. Math. 63, 401-427 (2005) · doi:10.1090/S0033-569X-05-00961-8
[8] Guo, L., Pan, L., Yin, G.: The perturbed Riemann problem and delta contact discontinuity in chromatography equations. Nonlinear Anal. 106, 110-123 (2014) · Zbl 1295.35321 · doi:10.1016/j.na.2014.04.016
[9] Guo, L., Sheng, W., Zhang, T.: The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system. Commun. Pure Appl. Anal. 9, 431-458 (2010) · Zbl 1197.35164 · doi:10.3934/cpaa.2010.9.431
[10] Huang, F., Wang, Z.: Well-posedness for pressureless flow. Commun. Math. Phys. 222, 117-146 (2001) · Zbl 0988.35112 · doi:10.1007/s002200100506
[11] Keyfitz, B.L., Kranzer, H.C.: A system of nonstrictly hyperbolic conservation laws arising in elasticity. Arch. Ration. Mech. Anal. 72, 219-241 (1980) · Zbl 0434.73019 · doi:10.1007/BF00281590
[12] Korchinski, D.J.: Solution of a Riemann problem for a system of conservation laws possessing no classical weak solution. Thesis, Adelphi University (1977)
[13] Lai, G., Sheng, W., Zheng, Y.: Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions. Discrete Contin. Dyn. Syst. 31, 489-523 (2011) · Zbl 1222.35121 · doi:10.3934/dcds.2011.31.489
[14] Li, J., Zhang, T., Yang, S.: The Two-Dimensional Riemann Problem in Gas Dynamics. Pitman Monographs, vol. 98. Longman, Harlow (1998) · Zbl 0935.76002
[15] Lu, Y.G.: Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type. Nonlinear Anal., Real World Appl. 13, 235-240 (2012) · Zbl 1238.35058 · doi:10.1016/j.nonrwa.2011.07.029
[16] Mazzotti, M.: Nonclassical composition fronts in nonlinear chromatography: delta-shock. Ind. Eng. Chem. Res. 48, 7733-7752 (2009) · doi:10.1021/ie9001537
[17] Mazzotti, M., Tarafder, A., Cornel, J., Gritti, F., Guiochond, G.: Experimental evidence of a delta-shock in nonlinear chromatography. J. Chromatogr. A 1217, 2002-2012 (2010) · doi:10.1016/j.chroma.2010.01.059
[18] Nedeljkov, M., Oberguggenberger, M.: Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J. Math. Anal. Appl. 344, 1143-1157 (2008) · Zbl 1155.35059 · doi:10.1016/j.jmaa.2008.03.040
[19] Nedeljkov, M.: Shadow waves: entropies and interactions for delta and singular shocks. Arch. Ration. Mech. Anal. 197, 489-537 (2010) · Zbl 1201.35134 · doi:10.1007/s00205-009-0281-2
[20] Panov, E., Shelkovich, V.M.: δ’-shock waves as a new type of solutions to system of conservation laws. J. Differ. Equ. 228, 49-86 (2006) · Zbl 1108.35116 · doi:10.1016/j.jde.2006.04.004
[21] Rykov, Yu.G., Sinai, Ya.G., Weinan, E.: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177, 349-380 (1996) · Zbl 0852.35097 · doi:10.1007/BF02101897
[22] Shao, Z.: Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations. Z. Angew. Math. Phys. 67, 66 (2016) · Zbl 1355.35139 · doi:10.1007/s00033-016-0663-x
[23] Shao, Z.: The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation. Z. Angew. Math. Phys. 69, 44 (2018) · Zbl 1392.35218 · doi:10.1007/s00033-018-0937-6
[24] Shelkovich, V. M., One class of systems of conservation laws admitting delta-shocks, No. 2, 667-674 (2012) · Zbl 1291.35133 · doi:10.1142/9789814417099_0070
[25] Shen, C.: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IMA J. Appl. Math. 81, 76-99 (2016) · Zbl 1336.35281
[26] Shen, C.: The Riemann problem for the Chaplygin gas equations with a source term. Z. Angew. Math. Mech. 96, 681-695 (2016) · doi:10.1002/zamm.201500015
[27] Shen, C.: Delta shock wave solution for a symmetric Keyfitz-Kranzer system. Appl. Math. Lett. 77, 35-43 (2018) · Zbl 1380.35126 · doi:10.1016/j.aml.2017.09.016
[28] Shen, C., Sun, M.: Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J. Differ. Equ. 249, 3024-3051 (2010) · Zbl 1211.35193 · doi:10.1016/j.jde.2010.09.004
[29] Sheng, W., Zhang, T.: The Riemann problem for the transportation equations in gas dynamics. Mem. Am. Math. Soc. 137, 654 (1999) · Zbl 0913.35082
[30] Sun, M.: Interaction of elementary waves for the Aw-Rascle model. SIAM J. Appl. Math. 69, 1542-1558 (2009) · Zbl 1184.35208 · doi:10.1137/080731402
[31] Sun, M.: Singular solutions to the Riemann problem for a macroscopic production model. Z. Angew. Math. Mech. 97, 916-931 (2017) · doi:10.1002/zamm.201600171
[32] Tan, D., Zhang, T., Zheng, Y.: Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differ. Equ. 112, 1-32 (1994) · Zbl 0804.35077 · doi:10.1006/jdeq.1994.1093
[33] Temple, B.: Global solution of the Cauchy problem for a class of 2×\(22\times 2\) nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3, 335-375 (1982) · Zbl 0508.76107 · doi:10.1016/S0196-8858(82)80010-9
[34] Temple, B.: Systems of conservation laws with invariant submanifolds. Trans. Am. Math. Soc. 280, 781-795 (1983) · Zbl 0559.35046 · doi:10.1090/S0002-9947-1983-0716850-2
[35] Wang, G.: One-dimensional nonlinear chromatography system and delta-shock waves. Z. Angew. Math. Phys. 64, 1451-1469 (2013) · Zbl 1284.35284 · doi:10.1007/s00033-013-0300-x
[36] Wang, G.: The Riemann problem for one dimensional generalized Chaplygin gas dynamics. J. Math. Anal. Appl. 403, 434-450 (2013) · Zbl 1426.76207 · doi:10.1016/j.jmaa.2013.02.026
[37] Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159, 447-484 (1999) · Zbl 0948.35079 · doi:10.1006/jdeq.1999.3629
[38] Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252, 5951-5993 (2012) · Zbl 1248.35127 · doi:10.1016/j.jde.2012.02.015
[39] Yang, H., Zhang, Y.: Delta shock waves with Dirac delta function in both components for systems of conservation laws. J. Differ. Equ. 257, 4369-4402 (2014) · Zbl 1304.35422 · doi:10.1016/j.jde.2014.08.009
[40] Zhang, Q.: Interactions of delta shock waves and stability of Riemann solutions for nonlinear chromatography equations. Z. Angew. Math. Phys. 67, 1-14 (2016) · Zbl 1342.35265 · doi:10.1007/s00033-015-0604-0
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