Initial boundary value problem for a system derived from Eulerian droplet model for air particle flow

In this work, we study the initial boundary value problem for a non-strictly hyperbolic $2\times2$ system of equations in the quarter plane $x>0,t>0$ which is derived from Eulerian droplet model for air particle flow for velocity and volume fraction. We show the existence of weak asymptotic solutions to the initial value problem to the system using a regularisation, by a vanishing viscosity method when the initial velocity is bounded measurable, the initial volume fraction is integrable and the boundary data are bounded measurable. Here we use a generalization of the Hopf-Cole transformation. We also derive an explicit formula for the weak solution when the initial data are functions of bounded variation, the boundary datas are bounded and locally in the class of Lipschitz continuous functions. This construction involves the Hopf-Lax formula for the boundary value problem for the Burgers equation and the product of a bounded variation function with derivative of another bounded variation function using non-conservative Volpert product.