On an L_p approach to the stationary and nonstationary problems of the Ginzburg-Landau-Maxwell equations

In this paper, we study a stationary and a nonstationary problem of the Ginzburg-Landau-Maxwell equations with Coulomb gauge in the L_p framework. First we prove a unique existence of stationary solution near the constant state with a small external magnetic field. Moreover, we prove a globally in time existence of solutions to the time dependent Ginzburg-Landau-Maxwell equations with small initial data and external magnetic field, and we show its convergence to the corresponding stationary solution when time tends to infinity. The key of our approach is to use various L_p-L_q estimates of the analytic semigroup generated by the linearized problem. Especially our initial data belong to L₃ without any additional regularity.