Abstract
The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter \({\varepsilon}\) tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.
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Ambrosio L., Crippa G., Figalli A., Spinolo L.A.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J. Math. Anal. 41, 1890–1920 (2009)
Ancona F., Goatin P.: Uniqueness and stability of L ∞ solutions for Temple class systems with boundary and properties of the attenaible sets. SIAM J. Math. Anal. 34, 28–63 (2002)
Barti P., Bressan A.: The semigroup generated by a Temple class system with large data. Differ. Integr. Equ. 10, 401–418 (1997)
Bianchini S.: Stability of L ∞ solutions for hyperbolic systems with coinciding shocks and rarefactions. SIAM J. Math. Anal. 33, 959–981 (2001)
Bouchut, F.: On zero pressure gas dynamics. In: Perthame, B. (ed.) Advances in Kinetic Theory and Computing. Series on Advances in Mathematics for Applied Sciences, vol. 22, World Scientific Publishing, River Edge, NJ, pp. 171–190 (1994)
Bressan A., Goatin P.: Stability of L ∞ solutions of Temple class systems. Differ. Integr. Equ. 13, 1503–1528 (2000)
Chang, T., Hsiao, L.: The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41, Longman Scientific and Technical, Harlow (1989)
Chen G.Q., Liu H.: Formation of δ-shock and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isetropic fluids. SIAM J. Math. Anal. 34, 925–938 (2003)
Cheng H.J., Yang H.C.: Delta shock waves in chromatography equations. J. Math. Anal. Appl. 380, 475–485 (2011)
Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss.. Springer, Berlin (2000)
Dal Maso G., Lefloch P.G., Murat F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)
Danilov V.G., Shelkovich V.M.: Dynamics of progation and interaction of δ-shock waves in conservation law systems. J. Differ. Equ. 221, 333–381 (2005)
Danilov V.G., Shelkovich V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63(3), 401–427 (2005)
Hayes B.T., Lefloch P.G.: Measure solutions to a strictly hyperbolic system of conservation laws. Nonlinearity 9, 1547–1563 (1996)
Huang F., Wang Z.: Well-posedness for pressureless flow. Commun. Math. Phys. 222, 117–146 (2001)
Kalisch, H., Mitrovic, D.: Singular solutions of a fully nonlinear 2 × 2 system of conservation laws. http://arXiv:1105.4640v3 (2011)
Keyfitz B.L., Kranzer H.C.: Spaces of weighted measures for conservation laws with singular shock solutions. J. Differ. Equ. 118, 420–451 (1995)
Korchinski, D.J.: Solution of a Riemann problem for a system of conservation laws possessing noclassical weak solution. Thesis, Adelphi University (1977)
Li J.Q., Warnecke G.: Generalized characteristics and the uniqueness of entropy solutions to zero-pressure gas dynamics. Adv. Differ. Equ. 8, 961–1004 (2003)
Li J.Q., Yang H.C.: Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure flow in gas dynamics. Q. Appl. Math. 59, 315–342 (2001)
Li, J.Q., Zhang, T., Yang, S.L.: The Two-dimensional Riemann Problem in Gas Dynamics. Pitman Monographs 98, Longman, Harlow (1998)
Mazzotti, M.: Occurrence of a delta-shock in non-linear chromatography. Proc. Appl. Math. Mech. 7. doi:10.1002/pamm.200700912
Mazzotti M.: Nonclassical composition fronts in nonlinear chromatography: delta-shock. Ind. Eng. Chem. Res. 48, 7733–7752 (2009)
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)
Nedeljkov M.: Delta and singular delta locus for one dimensional systems of conservation laws. Math. Methods Appl. Sci. 27, 931–955 (2004)
Nedeljkov M.: Singular shock waves in interactions. Q. Appl. Math. 66, 281–302 (2008)
Nedeljkov M.: Shadow waves: entropies and interactions for delta and singular shocks. Arch. Ratio. Mech. Anal. 197(2), 489–537 (2010)
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)
Panov E.Y., Shelkovich V.M.: δ’-shock waves as a new type of solutions to system of conservation laws. J. Differ. Equ. 228, 49–86 (2006)
Serre D.: Solutions à à variations bornées pour certains systèmes hyperboliques de lois de conservation. J. Differ. Equ. 68(2), 137–168 (1987)
Serre, D.: Systems of Conservation Laws I/II. Cambridge University Press, Cambridge (1999)/(2000)
Shelkovich, V.M.: δ- and δ’-shock types of singular solutions to systems of conservation laws and the transport and concentration processes. Uspekhi Mat. Nauk 63:3(381), 73–146 (2008)
Shen C.: Wave interactions and stability of the Riemann solutions for the chromatography equations. J. Math. Anal. Appl. 365, 609–618 (2010)
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)
Sheng W., Zhang T.: The Riemann problem for the transportation equations in gas dynamics. Mem. Am. Math. Soc. 137, 654 (1999)
Smoller J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn.. Springer, New York (1994)
Sun M.: Delta Shock waves for the chromatography equations as self-similar viscosity limits. Q. Appl. Math. 69, 425–443 (2011)
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)
Temple B.: Systems of conservation laws with invariant submanifolds. Trans. Am. Math. Soc. 280(2), 781–795 (1983)
Shelkovich V.M.: The Riemann problem admitting δ−, δ’-shocks and vacuum states (the vanishing viscosity approach). J. Differ. Equ. 231, 459–500 (2006)
Weinan E., Rykov Y.G., Sinai Y.G.: Generalized variational principles, globalweak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177, 349–380 (1996)
Yang H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159, 447–484 (1999)
Zeldovich, Y.B., Myshkis, A.D.: Elements of Mathematical Physics: Medium Consisting of Noninteracting Particles (in Russian). Nauka, Moscow (1973)
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Research partially supported by the National Science Foundation of China (10971130), the Science Key Project of Education Department of Anhui Province (KJ2012A055) and the Doctoral Research Fund of Anhui University of Architecture (2011101-6).
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Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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Wang, G. One-dimensional nonlinear chromatography system and delta-shock waves. Z. Angew. Math. Phys. 64, 1451–1469 (2013). https://doi.org/10.1007/s00033-013-0300-x
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DOI: https://doi.org/10.1007/s00033-013-0300-x