Lifespan of the Non-resistive Hall-MHD System with Small Magnetic Gradient

TL;DR

This paper investigates the lifespan of solutions to the non-resistive Hall-MHD system, demonstrating that small initial magnetic gradients can lead to arbitrarily long solution lifespans, with explicit lower bounds provided.

Contribution

It establishes new lifespan estimates for the non-resistive Hall-MHD system based on initial magnetic gradient size, extending understanding of solution longevity.

Findings

Lifespan can be arbitrarily large with small initial magnetic gradients

Explicit lower bounds for lifespan are derived for both viscous and inviscid cases

Small initial magnetic gradients significantly influence solution longevity

Abstract

In this paper, we consider the non-resistive axially symmetric Hall-MHD system. We show that the lifespan of their strong solutions can be arbitrarily large if their initial magnetic gradient are small enough. Precise lifespan lower bounds for both viscid and inviscid cases are given.