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

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