Global existence and $L^{p}$ convergence rates of planar waves for three-dimensional bipolar Euler-Poisson systems

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. We show the global existence and $L^{p}$ convergence rates of planar diffusion waves for multi-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. A frequency decomposition and approximate Green function based on delicate energy method are used to get the optimal decay rates of the planar diffusion waves. To our knowledge, the $L^p(p\in[2,+\infty])$-convergence rate of planar waves improves the previous results about the $L^2$-convergence rates.