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

TL;DR

This paper establishes the global well-posedness and strong convergence of solutions for the compressible Euler-Maxwell system, introducing a novel dissipation characterization and error estimates at the relaxation limit.

Contribution

It provides the first global-in-time strong convergence result for the relaxation procedure with ill-prepared data, using a new dissipation structure analysis across frequency regimes.

Findings

Global well-posedness of classical solutions near equilibrium

Quantitative error estimates of order O(ε) between Euler-Maxwell and drift-diffusion models

New dissipation characterization using frequency space partitioning

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 show the global well-posedness of classical solutions being a sharp small perturbation of constant equilibrium in a critical regularity setting, uniformly with respect to the relaxation parameter $\varepsilon>0$. Then, for all times $t>0$, we derive quantitative error estimates at the rate $O(\varepsilon)$ between the rescaled Euler-Maxwell system and the limit drift-diffusion model. To the best of our knowledge, this work provides the first global-in-time strong convergence for the relaxation procedure in the case of ill-prepared data. In order to prove our results, we develop a new characterization of the dissipation structure for the linearized Euler-Maxwell system with respect to the relaxation parameter $\varepsilon$. This is done by partitioning the frequency space into three distinct regimes: low, medium and high frequencies, each associated with a different behaviour of the solution. Then, in each regime, the use of efficient unknowns and Lyapunov functionals based on the hypocoercivity theory leads to uniform a priori estimates.