Euler-Maxwell systems are fluid models arising in plasma physics. In both isentropic and non-isentropic cases, such systems
admit non-constant steady-state solutions with zero velocity. For the Cauchy problem or the periodic problem with initial data near the steady-states, we show global existence and the convergence of smooth solutions toward these states as the time goes to infinity. In the proof of the above result, we mainly use three techniques to yield energy estimates. These techniques are the choice of symmetrizer of the systems, the existence of anti-symmetric matrices and an induction argument on the order of space-time derivatives of solutions.