Eulerian droplet model: Delta-shock waves and solution of the Riemann problem

摘要

We study an Eulerian droplet model which can be seen as the pressureless gassystem with a zeroth order linear source term. This leads to the analysis of asubsystem of this model and the inviscid Burgers equation with a zeroth orderlinear source term. A condition for the loss of regularity of the solution tothe Burgers equation with source term is established, which condition appliesto the droplet model. We show that the characteristics are no longer straightlines due to the presence of the source term but tend asymptotically tostraight lines. The Riemann problem for the droplet model is constructivelysolved by going through the solution of the Riemann problems for the inviscidBurgers equation with a zeroth order linear source term and the subsystem,respectively. The existence of an entropic solution to the generalizedRankine-Hugoniot conditions which are given by a nonlinear system of ordinarydifferential equations, is proven for the Riemann solution. Some numericalillustrations are presented.