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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.06278 |
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| _version_ | 1866917585833951232 |
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| author | Jeong, In-Jee Oh, Sung-Jin |
| author_facet | Jeong, In-Jee Oh, Sung-Jin |
| contents | We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig--Ponce--Rolvung--Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola--Metcalfe--Tataru in the case of elliptic principal term.
A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig--Ponce--Rolvung--Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola--Metcalfe--Tataru to the present case of nonelliptic principal term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field Jeong, In-Jee Oh, Sung-Jin Analysis of PDEs Mathematical Physics We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig--Ponce--Rolvung--Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola--Metcalfe--Tataru in the case of elliptic principal term. A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig--Ponce--Rolvung--Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola--Metcalfe--Tataru to the present case of nonelliptic principal term. |
| title | Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2402.06278 |