Existence results for the A – φ – B magnetodynamic formulation of the Maxwell system with skin and proximity effects

ABSTRACT

The $\mathbf{\text{A}}-\phi -\mathbf{\text{B}}$ magnetodynamic Maxwell system given in its potential and space-time formulation is a popular model considered in the engineering community. It allows to model some phenomena such as eddy current losses in multiple turn winding. Indeed, in some cases, they can significantly alter the performance of the devices, and consequently can no more be neglegted. It turns out that this model is not yet analytically studied, therefore we here consider its well-posedness. First, the existence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems is established. Second, using energy estimates, existence and uniqueness of the weak solution of the $\mathbf{\text{A}}-\phi -\mathbf{\text{B}}$ is deduced.

Keywords:

• Maxwell equations
• potential formulation
• degenerate parabolic problems
• Eddy current losses

AMS (MOS) subject classifications:

• 35H65
• 35Q60
• 78A25

Disclosure statement

No potential conflict of interest was reported by the authors.