On the regularity of solutions of a 3D coupled parabolic-elliptic equations via partial components

A famous heuristic theory in hydrodynamics named magnetic relaxation argues that the solutions of magnetic relaxation equations are conjectured to converge towards Euler equilibria as time tends to infinite. In this paper, we consider the regularity of solutions of a 3D coupled parabolic-elliptic equations arising in the magnetic relaxation theory. It is firstly shown that the solutions of such parabolic-elliptic model obey a regularity criterion which involves one velocity, one current density and two components of gradient tensor of the magnetic field. Then, under some suitable assumptions on indices of integrable spaces, several improved versions of this regularity criterion are given. Our results might have potential applications towards a rigorous theory of magnetic relaxation.