Global Solvability for a Nonlinear Semi-static Maxwell's Equation (Journal of Partial Differential Equations, 2006)
In this paper we study a nonlinear Maxwell's system in a highly conductive medium in which the displacement current is neglected. The magnetic field H satisfies a quasilinear evolution system: H_t + ∇ × [r(x, t, |H|, |∇ × H|)∇ × H] = F(x, t,H), where the resistivity r is assumed to depend upon the strengths of electric and magnetic fields while the internal magnetic current F depends upon the magnetic field. It is shown that under appropriate structure conditions for r and F the above nonlinear system subject to appropriate initial-boundary conditions has a unique global solution.
Keywords: Nonlinear Maxwell's Equations; Bean-like critical-state model; Global existence and uniqueness