Global solutions to Maxwell equations in a ferromagnetic medium
The Cauchy problem for the Maxwell system with the separated magnetization \(M\) \[ \partial_tE-\text{curl} H=0,\quad \partial_tH+\text{curl} E=0,\quad \partial_tM=F(M,H), \] \[ \text{div} E=0,\quad \text{div} (H+M)=0 \] is studied. The nonlinear term \(F(M,H)\) corresponds to the Landau-Lifschitz model for the propagation of the electromagnetic field in a ferromagnetic medium. The aim of the paper is to prove the existence of global finite energy solutions. The distribution \((E,H,M)\) is a finite energy solution in \(\Omega_T=[0,T]\times \mathbb{R}^3\) if all its components belong to \(C^0([0,T]; L^2(\mathbb{R}^3))\) and \(M\) is in \(L^{\infty}([0,T]\times \mathbb{R}^3)\). The uniqueness and regularity properties of these solutions are studied also. Particularly, it is shown that if the initial data are smooth, then the solution of the Cauchy problem remains smooth for all time. By suitable assumptions on \(F\) it is proved that if \(E_0, {H_0}_{\perp}\) belong to \(H^{\mu}, \mu\in]0,1]\), then \(E\) and \(H_{\perp}\) belong to \(C^0([0,+\infty[; H^{\mu}(\mathbb{R}^3))\).
scientific article; zbMATH DE number 1540531