Electromagnetic Processes in Doubly-Nonlinear Composites

Electromagnetic processes in inhomogeneous conductors are here described by coupling the Maxwell equations with nonlinear constitutive relations of the form B = B H x and J = J E H x , neglecting hysteresis and displacement currents. The latter equality may also account for the Hall effect. A doubly-nonlinear parabolic-hyperbolic equation is formulated, and existence of a solution is proved via approximation by time-discretization, derivation of a priori estimates, and passage to the limit via compensated compactness and compactness by strict convexity. It is then assumed that the medium is a composite that exhibits periodic oscillations in space. Convergence to a corresponding homogenized two-scale problem is proved as the oscillation period vanishes, via Nguetseng’s notion of twoscale convergence. Finally this two-scale formulation is proved to be equivalent to a coarse-scale problem.