Singular shock waves in interactions
Author:
Marko Nedeljkov
Journal:
Quart. Appl. Math. 66 (2008), 281-302
MSC (2000):
Primary 35L65, 35L67
DOI:
https://doi.org/10.1090/S0033-569X-08-01109-5
Published electronically:
February 7, 2008
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Abstract:
In a number of papers it has been shown that there exist one-dimensional systems such that they contain solutions with so-called overcompressive singular shock waves besides the usual elementary waves (shock and rarefaction waves as well as contact discontinuities).
One can see their definition for a general 2 $\times$ 2 system with fluxes linear in one of the dependent variables in Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws, Math. Method Appl. Sci. 27 (2004), 931–955. This paper is devoted to examining their interactions with themselves and elementary waves. After a discussion of systems given in a general form, a complete analysis will be given for the ion-acoustic system given in Keyfitz and Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differ. Equations 118 (1995), no. 2, 420–451.
References
- Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR 1763936
- V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of $\delta $-shock waves in conservation law systems, J. Differential Equations 211 (2005), no. 2, 333–381. MR 2125546, DOI https://doi.org/10.1016/j.jde.2004.12.011
- V. G. Danilov and V. M. Shelkovich, Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63 (2005), no. 3, 401–427. MR 2169026, DOI https://doi.org/10.1090/S0033-569X-05-00961-8
- Feimin Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations 30 (2005), no. 1-3, 283–304. MR 2131055, DOI https://doi.org/10.1081/PDE-200050026
- Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
- Marko Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Methods Appl. Sci. 27 (2004), no. 8, 931–955. MR 2055283, DOI https://doi.org/10.1002/mma.480
- M. Nedeljkov and M. Oberguggenberger, Delta shock wave and interactions in a simple model case, Submitted.
- De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
References
- A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, New York, 2000. MR 1816648 (2002d:35002)
- C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Heidelberg, 2000. MR 1763936 (2001m:35212)
- V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of shock waves in conservation law systems, J. Differ. Equations 211 (2005), 333-381. MR 2125546 (2006f:35173)
- ---, Delta-shock wave type solution of hyperbolic systems of conservation laws, Q. Appl. Math. 29 (2005), 401-427. MR 2169026 (2006j:35158)
- F. Huang, Weak solution to pressureless type system, Comm. Partial Differential Equations 30 (2005), no. 1-3, 283-304. MR 2131055 (2005k:35263)
- B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differ. Equations 118 (1995), no. 2, 420-451. MR 1330835 (96b:35138)
- P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973. MR 0350216 (50:2709)
- M. Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws, Math. Method Appl. Sci. 27 (2004), 931-955. MR 2055283 (2005g:35210)
- M. Nedeljkov and M. Oberguggenberger, Delta shock wave and interactions in a simple model case, Submitted.
- Tan, D., Zhang, T. and Zheng, Y., Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equations 112 (1994), 1-32. MR 1287550 (95g:35124)
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Additional Information
Marko Nedeljkov
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Yugoslavia
Email:
markonne@uns.ns.ac.yu, marko@im.ns.ac.yu
Keywords:
conservation law systems,
singular shock wave,
interaction of singularities
Received by editor(s):
June 10, 2006
Published electronically:
February 7, 2008
Additional Notes:
The work is supported by Serbian Ministry of Science and Enviroment Protection, Grant No. 144016
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.
