Publications
Home
Schools
Mathematics
Publications
- Title
- Wellposedness of the Electron MHD Without Resistivity for Large Perturbations of the Uniform Magnetic Field
- KIAS Author
- Oh, Sung-Jin,Oh, Sung-Jin
- Journal
- ANNALS OF PDE, 2025
- Archive
-
- Abstract
- 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 Schr & ouml;dinger 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.