Asymptotic Limit Of Bipolar Steady-state Hydrodynamic Model In Radial Symmetric Form

In this paper,we consider the asymptotic limit of the bipolar steady-state hydrodynamic model with radial symmetry,including the zero-electron-mass limit,combined zero-electron-mass and zero-hole-mass limit,the relaxation time limit.The main methods and results are as follows:Firstly,we define the radial symmetric solutions and derive the bipolar steady-state hydrodynamic equations with appropriate boundary conditions.When the spatial dimension is greater than one,we use the Schauder fixed point theorem and the classical theory of second-order linear elliptic equation to get the solvability of the model under the appropriate small hypothesis.Moreover,we prove the uniqueness of the solution by using the method of energy estimation.Secondly,on this basis,we study the three kinds of limits which are mentioned above.First of all,we use the perturbation theory to expand the radial symmetric solutions in terms of specific parameters.We also derive the equations satisfied by any order approximate solutions.Then,we make the system parameters tend to zero,Through refined energy estimation,we give the error estimates of the radial symmetric solutions to the zero order approximate solutions,that is the strong convergence and convergence rate of the corresponding asymptotic limit.