On 3D Hall-MHD equations with fractional Laplacians: global well-posedness

TL;DR

This paper establishes the global well-posedness of the 3D Hall-MHD equations with fractional Laplacians for both small and certain large initial data, using advanced energy estimates and harmonic analysis techniques.

Contribution

It proves the global existence and uniqueness of solutions for the 3D Hall-MHD system with fractional Laplacians, including a new class of large-energy initial data.

Findings

Global well-posedness for small-energy solutions in $H^s$, $s>rac{5}{2}$

Construction of a special class of large-energy initial data with global solutions

Development of new energy bounds overcoming derivative loss in magnetic field

Abstract

Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians is studied. First, global well-posedness of small-energy solutions with general initial data in $H^s$, $s>\frac{5}{2}$, is proved. Second, a special class of large-energy initial data is constructed, with which the Cauchy problem is globally well-posed. The proofs rely upon a new global bound of energy estimates involving Littlewood-Paley decomposition and Sobolev inequalities, which enables one to overcome the $\frac{1}{2}$-order derivative loss of the magnetic field.