Preprint 2003-059

A Specific Hyperbolic System of Conservation Laws Admitting Delta-shock Wave Type Solutions

Abstract: We construct $\delta$-shock wave type solutions of the Cauchy problem for the system of conservation laws

u_t+(f(u)-v)_x=0,     v_t+g(u)_x=0,

where $f(u)$ and $g(u)$ are polynomials of degree $n$ and $n+1$, respectively, $n$ is even. A well known particular case of this system was studied in~\cite{KeKr},~\cite{KrKe} by B.~L.~Keyfitz and H.~C.~Kranzer. In this paper a techniques of the {\it weak asymptotics method} and the definition of a $\delta$-shock type solution introduced by V.~G.~Danilov and V.~M.~Shelkovich~\cite{DS3}--~\cite{DS5}, are used.

Geometric and physics sense of the Rankine--Hugoniot conditions for $\delta$-shocks is given for the above system, for the system

u_t+f(u)_x=0,     v_t+(g(u)v)_x=0,

and for the well-known zero-pressure gas dynamics system. The geometric aspect of $\delta$-shock wave formation from sufficiently smooth compactly supported initial data is considered. Namely, the construction for the position of $\delta$-shock in a breaking wave is given.

Paper:

Available as PDF (304 Kbytes) or gzipped PostScript (224 Kbytes; uncompress using gunzip).

Author(s):

V. M. Shelkovich, <shelkv@svm.abu.spb.ru>

Publishing information:

Comments:

Revised version submitted on December 2 2003.

Submitted by:

<shelkv@svm.abu.spb.ru> September 16 2003.