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

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.