Well-posedness Theory For A Generalization Of The Pressureless Euler-Poisson System

We consider a class of nonlinear hyperbolic systems, which generalize the Euler-Poisson system for the evolution of fluid flows without pressure term. We study the initial value problem and establish new results even for the Euler-Poisson system.We generalize here a method introduced by LeFloch in 1990 (based on the Volpert product and the Lax formula) and establish a well-posed theory when one component of the system is a measure-valued function while the second one has bounded variation. Existence is established for general initial data, while uniqueness is guaranteed only when the initial data does not generate rarefaction centers. We first solve a nonconservative version of the problem and construct solutions with bounded variation,. The solutions to the systems of interest is then obtained by differentiation, which provides us with a complete theory of existence and uniqueness for both formulations. Special care is needed to handle regions where one of the variables of the systems (the fluid density of the Euler-Poisson system) vanishes. However, this method cannot be generalized to the case more than one dimension.In order to handle with the N-dimensional case, we introduce a probability approach, constructing a sticky practical method with the help of which we find a process and give a solution of Euler-Poisson system. However, the method from the Theory of probability cannot treat the generalized system.