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

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.**A. Bressan,*Hyperbolic Systems of Conservation Laws*, Oxford University Press, New York, 2000. MR**1816648 (2002d:35002)****2.**C. Dafermos,*Hyperbolic Conservation Laws in Continuum Physics*, Springer-Verlag, Heidelberg, 2000. MR**1763936 (2001m:35212)****3.**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)****4.**-,*Delta-shock wave type solution of hyperbolic systems of conservation laws*, Q. Appl. Math.**29**(2005), 401-427. MR**2169026 (2006j:35158)****5.**F. Huang,*Weak solution to pressureless type system*, Comm. Partial Differential Equations**30**(2005), no. 1-3, 283-304. MR**2131055 (2005k:35263)****6.**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)****7.**P. D. Lax,*Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves*, SIAM, Philadelphia, 1973. MR**0350216 (50:2709)****8.**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)****9.**M. Nedeljkov and M. Oberguggenberger,*Delta shock wave and interactions in a simple model case*, Submitted.**10.**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

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.