Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911186574901248 |
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| author | Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun |
| author_facet | Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun |
| contents | We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of $\mathcal{O}(\varepsilon)$ between strong solutions of the relaxed Euler system and the porous medium equation in $\mathbb{R}^d$ ($d\geq1$) for \emph{ill-prepared} initial data. In a well-prepared setting, we derive an enhanced convergence rate of order $\mathcal{O}(\varepsilon^2)$ between the solutions of the relaxed compressible Euler system and their first-order asymptotic approximation. Regarding the relaxed Euler-Maxwell system, we prove the global strong convergence of its solutions to the drift-diffusion model in $\mathbb{R}^3$ in an \emph{ill-prepared} setting. These results are achieved by developing a new asymptotic expansion approach that, combined with stream function techniques, ensures uniform-in-time error estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04103 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces Crin-Barat, Timothée Peng, Yue-Jun Shou, Ling-Yun Analysis of PDEs We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of $\mathcal{O}(\varepsilon)$ between strong solutions of the relaxed Euler system and the porous medium equation in $\mathbb{R}^d$ ($d\geq1$) for \emph{ill-prepared} initial data. In a well-prepared setting, we derive an enhanced convergence rate of order $\mathcal{O}(\varepsilon^2)$ between the solutions of the relaxed compressible Euler system and their first-order asymptotic approximation. Regarding the relaxed Euler-Maxwell system, we prove the global strong convergence of its solutions to the drift-diffusion model in $\mathbb{R}^3$ in an \emph{ill-prepared} setting. These results are achieved by developing a new asymptotic expansion approach that, combined with stream function techniques, ensures uniform-in-time error estimates. |
| title | Global convergence rates in the relaxation limits for the compressible Euler and Euler-Maxwell systems in Sobolev spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.04103 |