Research On Weak Solutions Of Euler-Poisson Equations

In this paper,we mainly study the weak solution of the unipolar semiconductor hydrodynamic model with variable damping coefficient,and obtain the global existence of the cylindrically symmetric solutions of the isentropic Euler-Poisson equations and the spherically symmetric solutions of the isothermal Euler-Poisson equations,as well as the large time behavior of the spherically symmetric weak entropy solutions of the isothermal Euler-Poisson equations.First of all,for the global existence of the cylindrically symmetric solution of the isentropic Euler-Poisson equations,the approximate solution is constructed by the viscous disappearance method,and the strong convergence of the approximate solution is obtained by using the compensating compactness method and the quasi-coupling method,thus completing the proof of the global existence of the cylindrically symmetric weak entropy solution of the system.Then,for the isothermal Euler-Poisson equations,the global existence of the spherically symmetric weak entropy solution of the system is obtained by using the viscous disappearance method and the compensated compactness method.Finally,on the basis of the global existence of the spherically symmetric solution of the isothermal Euler-Poisson equations,the large-time behavior of the spherically symmetric weak entropy solution of the isothermal Euler-Poisson equations is considered.Through the energy method and entropy estimation,it is proved that the weak entropy solution of the equations converges to the smooth solution of the corresponding steady-state equation.