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

Abstract

This paper is devoted to studying the well-posedness, conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell (NSM) equations with the standard Ohm's law in $\mathbb{R}^d$ for $d \in \{2,3\}$. More precisely, the global well-posedness is established in case of fractional Laplacian velocity $(-\Delta)^\alpha v$ with $\alpha = \frac{d}{2}$ for suitable data. In addition, the local well-posedness in the inviscid case is also provided for sufficient smooth data, which allows us to study the inviscid limit of associated positive viscosity solutions in the case $\alpha = 1$, where an explicit bound on the difference is given. Furthermore, in three dimensions if the initial data satisfies futher suitable conditions then magnetic helicity is conserved as the electric conductivity goes to infinity. On the other hand, in the case $\alpha = 0$ the stability near a magnetohydrostatic equilibrium with a constant (or equivalently bounded) magnetic field is also obtained in which nonhomogeneous Sobolev norms of the velocity and electric fields, and for $p \in (2,\infty]$ the $L^p$ norm of the magnetic field converge to zero as time goes to infinity with an implicit rate. In this velocity damping case, the situation is different both in case of the two and a half, and three-dimensional (Hall)-magnetohydrodynamics ((H)-MHD) system, where an explicit rate of convergence in infinite time is computed for both the velocity and magnetic fields in nonhomogeneous Sobolev norms. Therefore, it seems that there is a gap between NSM and MHD in terms of the norm convergence of the magnetic field and the rate of decaying in time, even the latter equations can be proved as a limiting system of the former one in the sense of distributions as the speed of light tends to infinity.