Local well-posedness for the two-and-a-half-dimensional EMHD system with split fractional dissipation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917515292049408 |
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| author | Peng, Qirui |
| author_facet | Peng, Qirui |
| contents | We study the $2\frac12$-dimensional electron magnetohydrodynamics (EMHD) system on $\mathbb T^2$ with componentwise fractional dissipation: $\partial_t a+a_yb_x-a_xb_y=-Λ^αa$ and $\partial_t b-a_yΔa_x+a_xΔa_y=-Λ^βb$, where $0<α,β<2$. This system is a $2\frac12$-dimensional reduction of the magnetic equation in Hall--MHD/EMHD under the ansatz $B=\nabla\times(ae_z)+be_z$. We prove local well-posedness for initial data $(a_0,b_0)\in H^{s+1}(\mathbb T^2)\times H^s(\mathbb T^2)$ with $s\geq 2-\varepsilon$, provided that $α+β>2$. Thus neither component is required to carry a full Laplacian dissipation; the smoothing effects of the two fractional dissipations can be combined to control the Hall nonlinearity. The proof is based on Littlewood--Paley energy estimates, commutator bounds, and cancellations between the leading low--high interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20845 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for the two-and-a-half-dimensional EMHD system with split fractional dissipation Peng, Qirui Analysis of PDEs We study the $2\frac12$-dimensional electron magnetohydrodynamics (EMHD) system on $\mathbb T^2$ with componentwise fractional dissipation: $\partial_t a+a_yb_x-a_xb_y=-Λ^αa$ and $\partial_t b-a_yΔa_x+a_xΔa_y=-Λ^βb$, where $0<α,β<2$. This system is a $2\frac12$-dimensional reduction of the magnetic equation in Hall--MHD/EMHD under the ansatz $B=\nabla\times(ae_z)+be_z$. We prove local well-posedness for initial data $(a_0,b_0)\in H^{s+1}(\mathbb T^2)\times H^s(\mathbb T^2)$ with $s\geq 2-\varepsilon$, provided that $α+β>2$. Thus neither component is required to carry a full Laplacian dissipation; the smoothing effects of the two fractional dissipations can be combined to control the Hall nonlinearity. The proof is based on Littlewood--Paley energy estimates, commutator bounds, and cancellations between the leading low--high interactions. |
| title | Local well-posedness for the two-and-a-half-dimensional EMHD system with split fractional dissipation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.20845 |