Interactions between two rarefaction waves for the pressure-gradient equations in the gas dynamics

In this paper, we consider the interactions of forward and backward centered rarefaction waves for pressure-gradient equations. For the Goursat problem developed on the domain of interactions, using characteristic analysis, it is proved that there never appears vacuum in this field when there is no vacuum on characteristic boundary. With the help of uniform a priori estimate, the existence and uniqueness of smooth solution is obtained, the results of interactions between two rarefaction waves are presented. Moreover, the numerical illustrations agree completely with the theoretical analysis.