Approximations of Euler-Maxwell systems by drift-diffusion equations through zero-relaxation limits near the non-constant equilibrium

Due to extreme difficulties in numerical simulations of Euler-Maxwell equations, which are caused by the highly complicated structures of the equations, this paper concerns the simplification of the Euler-Maxwell system through the zero-relaxation limit towards drift-diffusion equations with non-constant doping functions. We carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near non-constant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness arguments. Furthermore, we generalize the stream function method to the non-constant equilibrium case, and together with the anti-symmetric structure of the error system and an induction argument, we establish global-in-time error estimates between smooth solutions to the Euler-Maxwell system and those to the drift-diffusion system, which are bounded by some power of the relaxation parameter.