Singular solutions of a system of a non-strictly hyperbolic system

Systems of the first order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of solutions involve in many cases product of distributions. In this paper we study such a system derived from Eulerian droplet model for air particle flow. This is a 2 x 2 non - strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term and obtain a weak asymptotic solution with general initial data and also get solution in the Colombeau algebra. We also study the vanishing viscosity limit and show that this limit is a distribution solution. Further we study the large time asymptotic behaviour of the viscous system. This important system, is not very well studied due to complexities in the analysis. As far as we know the only work done on this system is for Riemann type of initial data. The significance of this paper is that we work on the system having general initial data and not just initial data of the Riemann type.