Abstract
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.
Highlights
In this note, we consider a bipolar hydrodynamic model (Euler-Poisson system) in one space dimension
Zhou and Li [4] and Tsuge [5] discussed the unique existence of the stationary solutions for the one-dimensional bipolar hydrodynamic model with proper boundary conditions
Theorems 1.1 and 1.2 show that the nonlinear diffusive phenomena is maintained in the bipolar Euler-Poisson system with the interaction of two particles and the additional electric field, which implies that this diffusion effect is essentially due to the friction of momentum relaxation
Summary
We consider a bipolar hydrodynamic model (Euler-Poisson system) in one space dimension. Zhou and Li [4] and Tsuge [5] discussed the unique existence of the stationary solutions for the one-dimensional bipolar hydrodynamic model with proper boundary conditions. Ali and Jüngel [10] studied the global smooth solutions of Cauchy problem for multidimensional hydrodynamic models for two-carrier plasma.
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