Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations with the Hall effect

TL;DR

This paper investigates the well-posedness, magnetic helicity conservation, inviscid limit, and stability of the generalized Navier-Stokes-Maxwell equations with Hall effect, providing new results on existence, limits, and conservation laws in various regimes.

Contribution

It establishes global well-posedness for small initial data, local well-posedness for large data, and conservation of magnetic helicity under certain conditions, advancing understanding of NSM with Hall effect.

Findings

Global well-posedness for small initial data in viscous case.

Conservation of magnetic helicity as electric conductivity tends to infinity.

Asymptotic stability around a constant magnetic field with velocity damping.

Abstract

This paper is devoted to studying the well-posedness, (conditional) conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell equations (NSM) under the Hall effect in two and three dimensions. More precisely, in the viscous case we prove the global well-posedness of NSM for small initial data, which allows us to establish a connection with either the Hall-magnetohydrodynamics (H-MHD) system as the speed of light tends to infinity or NSM without the Hall coefficient as this constant goes to zero. In addition, in the inviscid case the local well-posedness of NSM is also obtained for possibly large initial data. Moreover, under suitable conditions on the initial data and additional assumptions of solutions to NSM in three dimensions, the magnetic helicity is conserved as the electric conductivity goes to infinity. It is different to the case of the fractional H-MHD with critical fractional Laplacian exponents for both the velocity and magnetic fields, where the conservation of magnetic helicity can be provided for smooth initial data without any further conditions on the solution. Furthermore, the asymptotic stability of NSM around a constant magnetic field is established in the case of having a velocity damping term.