Exact solutions to the three-dimensional incompressible magnetohydrodynamics equations without viscosity

The incompressible magnetohydrodynamics (MHD) equations have been widely used to describe many physical systems in geophysics, astrophysics, cosmology and engineering. In this paper, we construct two types of exact global solutions with elementary functions to the three-dimensional incompressible MHD equations without viscosity. The first type of solutions is expressed by exponential functions that are nonstationary and correspond to a generalization of the well-known Arnold–Beltrami–Childress (ABC) flow for the three-dimensional MHD system. The second type of solutions has rational forms that are rotational and are similar to the ABC flow. Both types of solutions can exhibit interesting local behaviors with infinite energy. Under special parameter values, these solutions can be reduced to those of the incompressible Euler equations.