Huang, Feimin; Li, Yeping Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. (English) Zbl 1242.35050 Discrete Contin. Dyn. Syst. 24, No. 2, 455-470 (2009). Summary: 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 \(L^{\infty }\) 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 \(L^{2}(\mathbb R)\) 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 and R. Pan, Arch. Ration. Mech. Anal. 166, No. 4, 359–376 (2003; Zbl 1022.76042)] for compressible Euler equations with damping. As a by product, it is shown that the bounded \(L^{\infty }\) entropy solution of the bipolar hydrodynamic model converges to the entropy solution of Euler equations with damping as \(t\rightarrow\infty\). Cited in 1 ReviewCited in 40 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs 35Q05 Euler-Poisson-Darboux equations 35Q31 Euler equations Keywords:entropy solution; energy estimates; one space dimension; initial data with vacuum; similarity solution Citations:Zbl 1022.76042 PDFBibTeX XMLCite \textit{F. Huang} and \textit{Y. Li}, Discrete Contin. Dyn. Syst. 24, No. 2, 455--470 (2009; Zbl 1242.35050) Full Text: DOI