Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks

Abstract

For one-dimensional system of zero-pressure gas dynamics, we derive the balance laws describing mass, momentum and energy transport from the area outside the δ-shock wave front onto its front. It is proved that the total mass and momentum of area outside δ-shock wave front and the the δ-shock wave front are conserved, while the kinetic energy of the area outside the δ-shock wave front and total kinetic energy are decreasing quantities. It is also proved that if the kinetic energy is conserved then the Cauchy problem cannot have any nontrivial δ-shock wave type solution.

Keywords:

• zero-pressure gas dynamics
• δ-shocks
• the Rankine–Hugoniot conditions
• transportation and concentration processes

AMS Subject Classifications::

• Primary 35L65
• Secondary 35L67, 76L05

Acknowledgements

V.M. Shelkovich was supported in part by DFG Project 436 RUS 113/895 and by Analytical Departmental Special Program ‘The development of scientific potential of the Higher School’, Project 2.1.1/1399. The authors are greatly indebted to O.S. Rozanova for fruitful discussions.