On the self-similar solutions of the magneto-hydro-dynamic equations

Abstract

In this paper, we show that, for the three dimensional incompressible magneto-hydro-dynamic equations, there exists only trivial backward self-similar solution in Lp(M.3) for p ≥ 3, under some smallness assumption on either the kinetic energy of the self-similar solution related to the velocity field, or the magnetic field. Second, we construct a class of global unique forward self-similar solutions to the three-dimensional MHD equations with small initial data in some sense, being homogeneous of degree −1 and belonging to some Besov space, or the Lorentz space or pseudo-measure space, as motivated by the work in [5].