  In this paper, the magneto-heating coupling model is studied in details, with
turbulent convection zone and the flow field involved. Our main work is to
analyze the well-posed property of this model with the regularity techniques.
For the magnetic field, we consider the space $H_0(curl)\cap H(div_0)$ and for
the heat equation, we consider the space $H_0^1(\Omega)$. Then we present the
weak formulation of the coupled magneto-heating model and establish the
regularity problem. Using Roth's method, monotone theories of nonlinear
operator, weak convergence theories, we prove that the limits of the solutions
from Roth's method converge to the solutions of the regularity problem with
proper initial data. With the help of the spacial regularity technique, we
derive the results of the well-posedness of the original problems when the
regular parameter $\epsilon\longrightarrow 0$. Moreover, with additional
regularity assumption for both the magnetic field and temperature variable, we
prove the uniqueness of the solutions.
