Formula for Entropy solutions for the $1$–D Pressureless Euler–Poisson System: Well-posedness of Entropy Solutions and the Asymptotic Behaviors

We are concerned with the Cauchy problem for the one-dimensional pressureless Euler–Poisson system, which describes the dust stars with the density being a finite Radon measure. For this Cauchy problem, we introduce three generalized potentials to establish a representative formula for entropy solutions, and prove the uniqueness of entropy solutions via the variational principle and the method of generalized characteristics. Furthermore, we employ this newly derived formula to analyze the asymptotic behaviors of entropy solutions: For the initial data $(\rho_0,u_0)$ with finite Radon measure density $\rho_0(\not\equiv 0)$ and bounded velocity $u_0$, we prove that the entropy solutions always decay to a single $\delta$-shock by showing that any two $\delta$-shocks must coincide each other in a finite time; in particular, it is interesting that, for the initial density with a nonempty compact support, the entropy solution will turn into a $\delta$-shock wave within a finite time, after which this $\delta$-shock wave will propagate linearly despite the characteristics in general are parabolas.