Global existence and asymptotic behavior of smooth solutions for a bipolar Euler-Poisson system in the quarter plane
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China
Boundary Value Problems 2012, 2012:21 doi:10.1186/1687-2770-2012-21Published: 16 February 2012
In the article, a one-dimensional bipolar hydrodynamic model (Euler-Poisson system) in the quarter plane is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The global existence of smooth small solutions for the corresponding initial-boundary value problem is firstly shown. Next, the asymptotic behavior of the solutions towards the nonlinear diffusion waves, which are solutions of the corresponding nonlinear parabolic equation given by the related Darcy's law, is proven. Finally, the optimal convergence rates of the solutions towards the nonlinear diffusion waves are established. The proofs are completed from the energy methods and Fourier analysis. As far as we know, this is the first result about the optimal convergence rates of the solutions of the bipolar Euler-Poisson system with boundary effects towards the nonlinear diffusion waves.
Mathematics Subject Classification: 35M20; 35Q35; 76W05.