Mass, momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

Abstract

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.

Keywords:

• zero-pressure gas dynamics
• δ-shocks, the Rankine–Hugoniot conditions
• transportation and concentration processes of mass, momentum and energy

AMS Subject Classifications::

• Primary 35L65
• Secondary 35L67
• 76L05

Acknowledgements

Rozanova and Shelkovich were supported by the Analytical departmental special program ‘The development of scientific potential of the Higher School’, project 2.1.1/1399. Shelkovich was also supported in part by DFG Project 436 RUS 113/895.