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Uspekhi Mat. Nauk, 2008, Volume 63, Issue 3(381), Pages 73–146 (Mi umn9196)  

This article is cited in 40 scientific papers (total in 40 papers)

$\delta$- and $\delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes

V. M. Shelkovich

St. Petersburg State University of Architecture and Civil Engineering

Abstract: This is a survey of some results and problems connected with the theory of generalized solutions of quasi-linear conservation law systems which can admit delta-shaped singularities. They are the so-called $\delta$-shock wave type solutions and the recently introduced $\delta^{(n)}$-shock wave type solutions, $n=1,2,…$, which cannot be included in the classical Lax–Glimm theory. The case of $\delta$- and $\delta'$-shock waves is analyzed in detail. A specific analytical technique is developed to deal with such solutions. In order to define them, some special integral identities are introduced which extend the concept of weak solution, and the Rankine–Hugoniot conditions are derived. Solutions of Cauchy problems are constructed for some typical systems of conservation laws. Also investigated are multidimensional systems of conservation laws (in particular, zero-pressure gas dynamics systems) which admit $\delta$-shock wave type solutions. A geometric aspect of such solutions is considered: they are connected with transport and concentration processes, and the balance laws of transport of ‘volume’ and ‘area’ to $\delta$- and $\delta'$-shock fronts are derived for them. For a ‘zero-pressure gas dynamics’ system these laws are the mass and momentum transport laws. An algebraic aspect of these solutions is also considered: flux-functions are constructed for them which, being non-linear, are nevertheless uniquely defined Schwartz distributions. Thus, a singular solution of the Cauchy problem generates algebraic relations between its components (distributions).

DOI: https://doi.org/10.4213/rm9196

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English version:
Russian Mathematical Surveys, 2008, 63:3, 473–546

Bibliographic databases:

UDC: 517.9
MSC: Primary 35L65; Secondary 35L67, 76L05
Received: 09.02.2008

Citation: V. M. Shelkovich, “$\delta$- and $\delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes”, Uspekhi Mat. Nauk, 63:3(381) (2008), 73–146; Russian Math. Surveys, 63:3 (2008), 473–546

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    This publication is cited in the following articles:
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    5. Shen Ch., Sun M., “Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws”, Nonlinear Anal., 73:10 (2010), 3284–3294  crossref  mathscinet  zmath  isi  elib  scopus
    6. Shen Chun, “The limits of Riemann solutions to the isentropic magnetogasdynamics”, Appl. Math. Lett., 24:7 (2011), 1124–1129  crossref  mathscinet  zmath  isi  scopus
    7. Shen Chun, Sun Meina, Wang Zhen, “Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws”, Nonlinear Anal., 74:14 (2011), 4754–4770  crossref  mathscinet  zmath  isi  scopus
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    12. Shen Chun, “Structural stability of solutions to the Riemann problem for a scalar conservation law”, J. Math. Anal. Appl., 389:2 (2012), 1105–1116  crossref  mathscinet  zmath  isi  scopus
    13. M. Sun, “Non-selfsimilar solutions for a hyperbolic system of conservation laws in two space dimensions”, J. Math. Anal. Appl., 395:1 (2012), 86–102  crossref  mathscinet  zmath  isi  scopus
    14. Meina Sun, “Interactions of delta shock waves for the chromatography equations”, Applied Mathematics Letters, 26:6 (2013), 631  crossref  mathscinet  zmath  isi  scopus
    15. S. Albeverio, A. Yu. Khrennikov, S. V. Kozyrev, S. A. Vakulenko, I. V. Volovich, “In memory of Vladimir M. Shelkovich (1949–2013)”, P-Adic Num Ultrametr Anal Appl, 5:3 (2013), 242  crossref  mathscinet  zmath  scopus
    16. Meina Sun, “Formation of Delta Standing Wave for a Scalar Conservation Law with a Linear Flux Function Involving Discontinuous Coefficients”, Journal of Nonlinear Mathematical Physics, 20:2 (2013), 229  crossref  mathscinet  isi  scopus
    17. Xiumei Li, Chun Shen, “Viscous Regularization of Delta Shock Wave Solution for a Simplified Chromatography System”, Abstract and Applied Analysis, 2013 (2013), 1  crossref  mathscinet  isi  scopus
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    19. Wang G., “One-Dimensional Nonlinear Chromatography System and Delta-Shock Waves”, Z. Angew. Math. Phys., 64:5 (2013), 1451–1469  crossref  mathscinet  zmath  isi  elib  scopus
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    21. Lihui Guo, Ying Zhang, Gan Yin, “Interactions of delta shock waves for the Chaplygin gas equations with split delta functions”, Journal of Mathematical Analysis and Applications, 410:1 (2014), 190–201  crossref  mathscinet  zmath  isi  scopus
    22. Gan Yin, Kyungwoo Song, “Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas”, Journal of Mathematical Analysis and Applications, 411:2 (2014), 506–521  crossref  mathscinet  zmath  isi  scopus
    23. Chun Shen, “On a regularization of a scalar conservation law with discontinuous coefficients”, J. Math. Phys, 55:3 (2014), 031502  crossref  mathscinet  zmath  adsnasa  isi  scopus
    24. Lihui Guo, Gan Yin, “Limit of Riemann Solutions to the Nonsymmetric System of Keyfitz-Kranzer Type”, The Scientific World Journal, 2014 (2014), 1  crossref  zmath  isi  scopus
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    26. A.P.al Choudhury, K. T. Joseph, M.R.. Sahoo, “Spherically symmetric solutions of multidimensional zero-pressure gas dynamics system”, J. Hyper. Differential Equations, 11:02 (2014), 269  crossref  mathscinet  zmath  isi  scopus
    27. Lihui Guo, Ying Zhang, Gan Yin, “Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions”, Math. Meth. Appl. Sci, 2014, n/a  crossref  mathscinet  isi  scopus
    28. M. Colombeau, “Irregular shock waves formation as continuation of analytic solutions”, Applicable Analysis, 2014, 1  crossref  mathscinet  isi  scopus
    29. Lihui Guo, Gan Yin, “The Riemann Problem with Delta Initial Data for the One-Dimensional Transport Equations”, Bull. Malays. Math. Sci. Soc, 2014  crossref  mathscinet  isi  scopus
    30. R.S.. Baty, “Modern infinitesimals and delta-function perturbations of a contact discontinuity”, International Journal of Aeroacoustics, 14:1 (2015), 25  crossref  mathscinet  isi  scopus
    31. M. Colombeau, “Weak asymptotic methods for 3-D self-gravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear Euler-Poisson equations”, J. Math. Phys, 56:6 (2015), 061506  crossref  mathscinet  zmath  isi
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    34. Shen Ch., “Riemann Problem For a Two-Dimensional Quasilinear Hyperbolic System”, Electron. J. Differ. Equ., 2015  mathscinet  isi
    35. Sun M., “Structural Stability of Solutions To the Riemann Problem For a Non-Strictly Hyperbolic System With Flux Approximation”, Electron. J. Differ. Equ., 2016, 126  mathscinet  isi  elib
    36. Abreu E., Colombeau M., Panov E., “Weak asymptotic methods for scalar equations and systems”, J. Math. Anal. Appl., 444:2 (2016), 1203–1232  crossref  mathscinet  zmath  isi  scopus
    37. Shao Zh., Huang M., “Interactions of Delta Shock Waves For the Aw-Rascle Traffic Model With Split Delta Functions”, J. Appl. Anal. Comput., 7:1 (2017), 119–133  crossref  mathscinet  isi  elib  scopus
    38. Abreu E., Colombeau M., Panov E.Yu., “Approximation of Entropy Solutions to Degenerate Nonlinear Parabolic Equations”, Z. Angew. Math. Phys., 68:6 (2017), 133  crossref  mathscinet  zmath  isi  scopus
    39. Guo L., Li T., Pan L., Han X., “The Riemann Problem With Delta Initial Data For the One-Dimensional Chaplygin Gas Equations With a Source Term”, Nonlinear Anal.-Real World Appl., 41 (2018), 588–606  crossref  mathscinet  zmath  isi  scopus
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