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.



Paper:
Available as PDF (320 Kbytes), Postscript (608 Kbytes) or gzipped PostScript (256 Kbytes; uncompress using gunzip).
Author(s):
E. Yu. Panov, <pey@novsu.ac.ru>
V. M. Shelkovich, <February 7>
Publishing information:
Comments:
Submitted by:
<shelkv@vs1567.spb.edu> March 15 2005.


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