Preprint 2005-012

δn-Shock Wave as a New Type Solutions of Hyperbolic Systems of Conservation Laws

E. Yu. Panov and V. M. Shelkovich

Abstract: A concept of a new type of singular solutions to hyperbolic systems of conservation laws is introduced. It is so-called δn-shock wave, where δn is n-th derivative of the delta function.

We introduce a definition of δ'-shock wave type solution for the system

ut+ f(u)x=0,    vt+(f'(u)v)x=0,    wt+(f''(u)v2+f'(u)w)x=0.
Within the framework of this definition, the Rankine--Hugoniot conditions for δ'-shock are derived and analyzed from geometrical point of view. We prove δ'-shock balance relations connected with area transportation. A solitary δ'-shock wave type solution to the Cauchy problem of the system of conservation laws
ut+(u2)x=0,    vt+2(uv)x=0,    wt+2(v2+uw)x=0
with piecewise continuous initial data is constructed.

These results show that solutions of hyperbolic systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.

Available as PDF (320 Kbytes), Postscript (608 Kbytes) or gzipped PostScript (256 Kbytes; uncompress using gunzip).
E. Yu. Panov, <>
V. M. Shelkovich, <February 7>
Publishing information:
Submitted by:
<> March 15 2005.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

Conservation Laws Preprint Server <>
2005-02-15 13:36:49 UTC