Analysis of a non-uniformly elliptic and nonlinear coupled parabolic–elliptic system arising in steel hardening

Abstract

The goal of this work is to analyse the existence of weak solutions to a coupled nonlinear parabolic-elliptic system derived from the heating industrial process of a steel workpiece, and whose unknowns are the electric potential, the magnetic vector potential and the temperature. We introduce the harmonic regime because of the different time scales related to the electric potential and the magnetic vector potential versus the temperature. This lead us to a new system of nonlinear partial differential equations.

Weak Solutions to a Nonuniformly Elliptic PDE System in the Harmonic Regime

“…The aim of this work is to analyze the existence of weak solutions to a nonlinear PDEs system arising in the heating induction-conduction process of a steel workpiece [7,8,10,11,13]. Since we are dealing with high oscillating sinusoidal in time for both electric potential and magnetic vector potential, we introduce a change of variables separating the two time scales.…”