Convergence rates in zero-relaxation limits for Euler-Maxwell and Euler-Poisson systems

Abstract It was proved that Euler-Maxwell systems converge globally-in-time to drift-diffusion systems in a slow time scaling, as the relaxation time goes to zero. The convergence was established to the Cauchy problem with smooth periodic initial data sufficiently close to constant equilibrium states. In this paper, we establish error estimates between smooth periodic solutions of Euler-Maxwell systems and those of drift-diffusion systems. Similar error estimates are also obtained for Euler-Poisson systems in place of Euler-Maxwell systems. The proof of these results uses stream function techniques together with energy estimates.