On the Well-Posedness and Long Time Behavior of the Hall-magnetohydrodynamics System

Abstract

We study the incompressible Hall-MHD system, an important model in plasma physics akin to the Navier-Stokes equations, using harmonic analysis tools. Chapter \ref{intro} consists of an introduction of the Hall-MHD system and its derivation from a two-fluid Euler-Maxwell system, along with a review of the mathematical preliminaries. Chapter \ref{w} concerns the well-posedness of the Hall-MHD system. For completeness, a proof of the global-in-time existence of the Leray-Hopf type weak solutions is included. In addition, we include a proof of the regularity criterion in \cite{D1}, which is of particular interest as it highlights the dissipation wavenumbers formulated via Littlewood-Paley theory. We then exploit the regularizing effect of diffusion and use a classical fixed point theorem to prove local-in-time existence of solutions to the generalized Hall-MHD system in certain Besov spaces as well as global-in-time existence of solutions to the hyper-dissipative electron MHD equations for small initial data in critical Besov spaces. Long time behaviour of solutions to the Hall-MHD system is studied in Chapter \ref{l}. We reproduce the proof of algebraic decay of weak solutions to the fully dissipative Hall-MHD system in \cite{CS1}; we then present our study of strong solutions to the Hall-MHD systems with mere one diffusion featuring the Fourier splitting technique. Under certain moderate assumptions, we show that the magnetic energy decays to $0$ and the kinetic energy converges to a certain constant in the resistive inviscid case, while the opposite happens in the viscous non-resistive case. Inspired by \cite{CDK}, we study the long time behaviour of solutions to the Hall-MHD system from the viewpoint of the determining Fourier modes. Via Littlewood-Paley theory, we formulate the determining wavenumbers, which bounds the low frequencies essential to the long time behaviour of the solutions. The fact that the determining wavenumbers can be estimated in a certain average sense suggests that the Hall-MHD system has finite degrees of freedom in a certain sense.