The uniform existence time and Zero-Alpha limit problem of the Euler-Poincaré equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916289103003648 |
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| author | Li, Min Yin, Zhaoyang |
| author_facet | Li, Min Yin, Zhaoyang |
| contents | We consider the Cauchy problem of the Euler-Poincaré equations in $\mathbb{R}^d$ with a varying dispersion parameter $α$. Based on the convex entropy structure and the modified commutator estimates, we have proved that the Euler-Poincaré equations have a uniform existence time with respect to $α$ in Sobolev spaces $H^s.$ Combined with the Bona-Simth method, we obtain convergence of the solutions to the Euler-Poincaré equations as $α\to 0$ in the same space where the initial data are located. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_15019 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The uniform existence time and Zero-Alpha limit problem of the Euler-Poincaré equations Li, Min Yin, Zhaoyang Analysis of PDEs 35Q35, 35Q51, 35L30 We consider the Cauchy problem of the Euler-Poincaré equations in $\mathbb{R}^d$ with a varying dispersion parameter $α$. Based on the convex entropy structure and the modified commutator estimates, we have proved that the Euler-Poincaré equations have a uniform existence time with respect to $α$ in Sobolev spaces $H^s.$ Combined with the Bona-Simth method, we obtain convergence of the solutions to the Euler-Poincaré equations as $α\to 0$ in the same space where the initial data are located. |
| title | The uniform existence time and Zero-Alpha limit problem of the Euler-Poincaré equations |
| topic | Analysis of PDEs 35Q35, 35Q51, 35L30 |
| url | https://arxiv.org/abs/2312.15019 |