A new characterization of the dissipation structure and the relaxation limit for the compressible Euler-Maxwell system
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866907962079969280 |
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| author | Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun Xu, Jiang |
| author_facet | Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun Xu, Jiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00277 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new characterization of the dissipation structure and the relaxation limit for the compressible Euler-Maxwell system Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun Xu, Jiang Analysis of PDEs 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. |
| title | A new characterization of the dissipation structure and the relaxation limit for the compressible Euler-Maxwell system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.00277 |