Singular shock waves in interactions

Author:
Marko Nedeljkov

Journal:
Quart. Appl. Math. **66** (2008), 281-302

MSC (2000):
Primary 35L65, 35L67

Published electronically:
February 7, 2008

MathSciNet review:
2416774

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Abstract | References | Similar Articles | Additional Information

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 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.

**1.**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****2.**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****3.**V. G. Danilov and V. M. Shelkovich,*Dynamics of propagation and interaction of 𝛿-shock waves in conservation law systems*, J. Differential Equations**211**(2005), no. 2, 333–381. MR**2125546**, 10.1016/j.jde.2004.12.011**4.**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**, 10.1090/S0033-569X-05-00961-8**5.**Feimin Huang,*Weak solution to pressureless type system*, Comm. Partial Differential Equations**30**(2005), no. 1-3, 283–304. MR**2131055**, 10.1081/PDE-200050026**6.**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**, 10.1006/jdeq.1995.1080**7.**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****8.**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**, 10.1002/mma.480**9.**M. Nedeljkov and M. Oberguggenberger,*Delta shock wave and interactions in a simple model case*, Submitted.**10.**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**, 10.1006/jdeq.1994.1093

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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

DOI:
https://doi.org/10.1090/S0033-569X-08-01109-5

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.