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

TL;DR

This paper investigates the mathematical solvability and well-posedness of a coupled magneto-heating model involving turbulent convection and flow fields, using advanced regularity and convergence techniques.

Contribution

It establishes the existence, regularity, and uniqueness of solutions for the magneto-heating coupling model with turbulent convection, extending the theoretical understanding of such complex systems.

Findings

Proved the well-posedness of the coupled model.

Established convergence of approximate solutions to true solutions.

Demonstrated uniqueness under additional regularity assumptions.

Abstract

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.