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In this paper, the magneto-heating coupling model is studied in details, withturbulent convection zone and the flow field involved. Our main work is toanalyze 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 forthe heat equation, we consider the space $H_0^1(\Omega)$. Then we present theweak formulation of the coupled magneto-heating model and establish theregularity problem. Using Roth's method, monotone theories of nonlinearoperator, weak convergence theories, we prove that the limits of the solutionsfrom Roth's method converge to the solutions of the regularity problem withproper initial data. With the help of the spacial regularity technique, wederive the results of the well-posedness of the original problems when theregular parameter $\epsilon\longrightarrow 0$. Moreover, with additionalregularity assumption for both the magnetic field and temperature variable, weprove the uniqueness of the solutions.

The Solvability Of Magneto-heating Coupling Model With Turbulent Convection Zone And The Flow Fields