Global existence and asymptotic behavior of smooth solutions for a bipolar Euler-Poisson system in the quarter plane

• Published in: Boundary value problems Vol. 2012; no. 1; pp. 1 - 13
• Main Author: Li, Yeping
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 16.02.2012; Hindawi Limited
• Subjects: Analysis; Approximations and Expansions; Asymptotic properties; Boundary value problems; Difference and Functional Equations; Diffusion waves; Fluid flow; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinearity; Ordinary Differential Equations; Partial Differential Equations; Planes; energy estimates; nonlinear diffusion waves; bipolar hydrodynamic model; smooth solutions
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/1687-2770-2012-21

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.