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

Abstract

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 $c\to \infty$. 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.