Existence and uniqueness of mild solutions to the magneto-hydro-dynamic equations

In this paper, we consider the incompressible magneto-hydro-dynamic equations in the whole space. We first show that there exist global mild solutions with small initial data in the scaling invariant space. The main technique we have used is implicit function theorem which yields necessarily continuous dependence of solutions for the initial data. Moreover, we gain the asymptotic stability of solutions as the time goes to infinity. Finally, as a byproduct of our construction of solutions in the weak Lp-spaces, the existence of self-similar solutions was established provided the initial data are small homogeneous functions.