Optimal decay rate of the bipolar Euler-Poisson system with damping in ${\mathbb{R}}^3$

Abstract

By rewriting a bipolar Euler-Poisson equations with damping into an Euler equation with damping coupled with an Euler-Poisson equation with damping, and using a new spectral analysis, we obtain the optimal decay results of the solutions in $L^2$-norm, which improve theose in \cite{Li3, Wu3}. More precisely, the velocities $u_1,u_2$ decay at the $L^2-$rate $(1+t{)}^{-54}$, which is faster than the normal $L^2-$rate $(1+t{)}^{-34}$ for the Heat equation and the Navier-Stokes equations. In addition, the disparity of two densities ${\rho }_1-{\rho }_2$ and the disparity of two velocities $u_1-u_2$ decay at the $L^2$-rate $(1+t{)}^{-2}$.

arXiv comment12 pages

References in corpus (1)

• Decay of the solution to the bipolar Euler-Poisson system with damping in ${\mathbb{R}}^3$