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

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.

arXiv comment27 pages. Welcome all comments

References in corpus (8)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
• Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations
• Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion
• Global well-posedness for Euler-Boussinesq system with critical dissipation
• Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation
• Global smooth solutions of MHD equations with large data
• Global large, smooth solutions of the 2D surface quasi-geostrophic equations