Vanishing viscosity solution to a 2×2$2 \times 2$ system of conservation laws with linear damping

Abstract Systems of the first‐order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of the solution involves, in many cases, the product of distributions. In this paper, we study such a system derived from Eulerian droplet model for air particle flow. This is a non‐strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term, obtain a weak asymptotic solution with general initial data and also get the solution in Colombeau algebra. We study the vanishing viscosity limit and show that this limit is a distributional solution. Further, we study the large‐time asymptotic behavior 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.