A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system

The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α−solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided and consequently the unique α−solution is obtained within a convenient class of distributions including the Dirac delta measure. It is shown that our constructed α−solution is reasonable compared with the known results using other methods. Furthermore, the result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, which enables us to see that the α−solution is not self-similar any more. It is shown that the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term.