Sung-Jin Oh : Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

Posted by Christof Sparber , part of the Analysis and Applied Mathematics Seminar.

At

April 15, 2024, 4 p.m.

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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 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.