The initial boundary value problem for inviscid barotropic fluid motion

In his paper [2], Ebin showed the local in time existence of solutions to the initial boundary value problem for inviscid barotropic fluid motion in a bounded domain provided that the initial velocity is subsonic and the initial density is nearly constant.The purpose of the present article is to prove without the above assumptions the existence and continuous dependence of solutions for the data.The inviscid barotropic fluid motion in a bounded domain \Omega\subset R^{\} with smooth boundary \partial\Omega is governed by the standard equations of fluid me- chanics ; \frac{dv}{dt}+\frac{p'(\rho)}{\rho}\nabla\rho=K (0. 1) in (0, T)\cross\Omega \frac{d\rho}{dt}+\rho div