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.

Keywords:

• steel hardening
• Maxwell's equations
• Joule's heating
• nonlinear parabolic–elliptic equations
• Sobolev spaces

2010 AMS Subject Classifications:

• 35A15
• 35G30
• 35J57
• 35K05
• 35K55
• 35Q61
• 35Q79
• 46E35

Acknowledgements

This research has been partially supported by the Project MTM2010–16401 of Ministerio de Economía y Competitivad with the participation of the FEDER, and Consejería de Educación y Ciencia de la Junta de Andalucía, research group FQM–315.