Global existence for Maxwell-Bloch systems

Maxwell-Bloch equations describe the propagation of an electromagnetic wave through a quantum medium. For any number of quantum levels, in space dimension 3, we show the global existence of weak (L2) solutions to the initial-value problem. In the case of smoother electromagnetic fields (with curl in L2), the solution is unique. For smooth data (Hs, s⩾2), the solutions remain smooth for all times.