On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical regularity spaces. First, we establish a global result for small initial data in the Besov spacesḂ 3 p −1 p,1 with 1 ≤ p < ∞, and the conservation of higher regularity. Second, in the case where the viscosity is equal to the magnetic resistivity, we obtain the global well-posedness for (small) initial data in the larger critical Besov spaces of typeḂ 1 2 2,r for any r ≥ 1. In the particular case r = 1, we also establish the local existence for large data, and supplement our results with continuation criteria. To the best of our knowledge, the present paper is the first one where well-posedness is proved for the Hall-MHD system, in a critical regularity setting