Diffusive relaxation limits of compressible Euler–Maxwell equations

This work is concerned with compressible Euler–Maxwell equations, which take the form of Euler equations for the conservation laws of mass density, current density and energy density for electrons, coupled to Maxwellʼs equations for self-consistent electromagnetic field. We give a model hierarchy of non-isentropic Euler–Maxwell equations from the point of view of diffusive relaxation limits. More precisely, inspired by Maxwell-type iteration, we construct new approximations and show that periodic initial-value problems of a certain scaled Euler–Maxwell equations have unique smooth solutions in a time interval independent of momentum relaxation time and energy relaxation time. Furthermore, it is proved that smooth solutions converge to solutions of drift-diffusion models and energy-transport models in the process of combined diffusive relaxation limits, and the corresponding convergence rates are also obtained.