Global-in-time error estimates of non-relativistic limits for Euler-Maxwell system near non-constant equilibrium

It was proved that Euler-Maxwell systems converge globally-in-time to Euler-Poisson systems near non-constant equilibrium states when the speed of light c -> infinity. In this paper, we establish the global-in-time error estimates between smooth solutions of Euler-Maxwell systems and those of Euler-Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field E, so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.