Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum

In this paper, a one-dimensional bipolar hydrodynamic model is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The large time behavior of L ^∞ entropy solutions of the bipolar hydrodynamic model is firstly studied. Previous works on this topic are mainly concerned with the smooth solution in which no vacuum occurs and the initial data is small. It is proved in this paper that any bounded entropy solution strongly converges to the similarity solution of the porous media equation or the heat equation in L ²(R) with time decay rate. The initial data can contain vacuum and can be arbitrarily large. The method is also applied to improve the convergence rate of [F.Huang, R.Pan, Arch. Rational Mech. Anal.,166(2003),359-376] for compressible Euler equations with damping. As a by product, it is shown that the bounded L ^∞ entropy solution of the bipolar hydrodynamic model converges to the entropy solution of Euler equations with damping as $t\rightarrow\infty$.