On the Well-posedness of Magnetohydrodynamics Equations for Incompressible Electrically-Conducting Fluids

Abstract

It is shown that the Cauchy problem of the equations in magnetohydrodynamics in the whole space is globally well-posed for any initial smooth and localized data. In general, the mathematical structure of solution shows that the coupled magnetic-vortical field has the characters of turbulence, if the initial data exceed certain size. In particular, if a strong magnetic force dominates the flow evolution, the current density possesses a cubic non-linearity. The solution of the non-linear problem has been constructed and has been expressed as an infinite series.