Local well-posedness for the Hall-MHD system in optimal Sobolev Spaces

Abstract

We show that the viscous resistive magneto-hydrodynamics system with Hall effect is locally well-posed in $H^s({\mathbb{R}}^n)\times H^{s+1-\epsilon }({\mathbb{R}}^n)$ with $s>\frac{n}{2}-1$ and any small enough $\epsilon >0$ such that $s+1-\epsilon >\frac{n}{2}$. This space is to date the largest local well-posedness space in the class of Sobolev spaces for the system. It is also optimal according to the predominant scalings of the two equations in the system.

arXiv commentReferences updated

References in corpus (1)

• On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations

Cited by in corpus (4)

• Non-unique weak solutions in Leray-Hopf class of the 3D Hall-MHD system
• Global Large Smooth Solutions for 3-D Hall-magnetohydrodynamics
• Anomalous dissipation of energy and magnetic helicity for the electron-MHD system
• Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics