On 3D Hall-MHD Equations with Fractional Laplacians: Global Well-Posedness

Abstract

The Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians (−Δ) 1 2 is studied. The well-posedness of 3D incompressible Hall-MHD equations remains an open problem with fractional diffusion (−Δ) 1 2. First, global well-posedness of small-energy solutions with general initial data in H s , s > 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 1 2 -order derivative loss of the magnetic field.

A class of global large, smooth solutions for the magnetohydrodynamics with Hall and ion‐slip effects

“…It is quite rare to prove the existence of large, smooth, global solutions for quasilnear system. Under a class of large initial data, we found some results for incompressible Navier-Stokes equations, the incompressible MHD equations, and the incompressible Hall-MHD equations; see previous studies 8,9,11,13,18,[24][25][26][27][28][29][30][31][32][33][34][35][36] for details. Those motivate us to study the global well-posedness of the Cauchy problem of Equation (1.1) with large initial data.…”