Non-unique solutions for electron MHD
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914807621353472 |
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| author | Dai, Mimi |
| author_facet | Dai, Mimi |
| contents | We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus $\mathbb T^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L^γ_tW^{1,p}_x$ for appropriate $(γ, p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $γ$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-unique solutions for electron MHD Dai, Mimi Analysis of PDEs 35Q35, 76B03, 76D09, 76E25, 76W05 We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus $\mathbb T^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L^γ_tW^{1,p}_x$ for appropriate $(γ, p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $γ$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity. |
| title | Non-unique solutions for electron MHD |
| topic | Analysis of PDEs 35Q35, 76B03, 76D09, 76E25, 76W05 |
| url | https://arxiv.org/abs/2405.14127 |