On the existence and temporal asymptotics of solutions for the two and half dimensional Hall MHD

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions for the 2.5-dimensional Hall MHD equations, establishing conditions for global solutions and decay rates, and analyzing the effects of fluid coupling.

Contribution

It provides new results on the existence, uniqueness, decay, and asymptotic profiles of solutions for the Hall MHD equations, including cases with and without fluid effects.

Findings

Global existence of strong solutions under small initial data

Decay rates of solutions and their asymptotic profiles

Existence of solutions locally in time and blow-up criteria

Abstract

In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking the velocity field $u$ and the magnetic field $B$ of the form $u(t,x,y)=\left(\nabla^{\perp}\phi(t,x,y), W(t,x,y)\right)$ and $B(t,x,y)=\left(\nabla^{\perp}\psi(t,x,y), Z(t,x,y)\right)$. We begin with the Hall equations (without the effect of the fluid part). We first show the long time behavior of weak solutions and weak-strong uniqueness. We then proceed to prove the existence of unique strong solutions locally in time and to derive a blow-up criterion. We also demonstrate that the strong solution exists globally in time and decay algebraically if some smallness conditions are imposed. We further improve the decay rates of $\psi$ using the structure of the equation of $\psi$. As a consequence of the decay rates of $(\psi,Z)$, we find the asymptotic profiles of $(\psi,Z)$. We finally show that a small perturbation of initial data near zero can be extended to a small perturbations near harmonic functions. In the presence of the fluid filed, the results, by comparison, fall short of the previous ones in the absence of the fluid part. We prove two results: the existence of unique strong solutions locally in time and a blow-up criterion, and the existence of unique strong solutions globally in time with some smallness condition on initial data.