On the Lifespan of Axisymmetric Hall-MHD with Swirl

TL;DR

This paper investigates the lifespan of solutions to the axisymmetric Hall-MHD system with swirl, providing explicit bounds and analyzing the effects of resistivity approaching zero.

Contribution

It establishes a lower bound on the lifespan of solutions based on initial swirl size and examines the lifespan behavior as resistivity diminishes.

Findings

Lifespan can be made arbitrarily large with small initial swirl.

Explicit lower bounds for solution lifespan are derived.

Lifespan behavior as resistivity tends to zero is analyzed.

Abstract

In this paper, we study the three-dimensional inviscid incompressible resistive Hall-MHD system in the axisymmetric setting with nontrivial swirl velocity and purely azimuthal magnetic. Assuming only that the swirl component of the initial velocity is sufficiently small, we prove that the lifespan of the strong solution can be sufficiently large. An explicit lifespan lower bound in terms of the size of the initial swirl is given. Moreover, we also study the behavior of the lifespan as the resistivity tends to zero.