A new characterization of the dissipation structure and the relaxation limit for the compressible Euler-Maxwell system

Abstract

We investigate the three-dimensional compressible Euler-Maxwell system, a model for simulating the transport of electrons interacting with propagating electromagnetic waves in semiconductor devices. First, we establish the global well-posedness of classical solutions near constant equilibrium in a critical regularity setting, uniformly with respect to the relaxation parameter $\epsilon >0$. Then, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate $O\left(\epsilon \right)$ between the rescaled Euler-Maxwell system and the limiting drift-diffusion model. This provides the first global-in-time strong convergence result for the relaxation procedure in the case of ill-prepared data so far.

We propose a new characterization of the dissipation structure for the non-symmetric relaxation of linearized Euler-Maxwell system, which partitions the frequency space into three distinct regimes (low, medium and high frequencies) associated with different behaviors of the solution. Within each regime, the application of Lyapunov functionals based on the hypocoercivity theory reveals the expected dissipative properties. Moreover, two correction functions are employed to take care of the initial layers in the relaxation convergence.