Relaxation Limit and Global Existence of Smooth Solutions of Compressible Euler–Maxwell Equations

We consider smooth periodic solutions for the Euler–Maxwell equations, which are a symmetrizable hyperbolic system of balance laws. We proved that as the relaxation time tends to zero, the Euler–Maxwell system converges to the drift-diffusion equations at least locally in time. The global existence of smooth solutions is established near a constant state with an asymptotic stability property.