GLOBAL WEAK SOLUTIONS AND ATTRACTORS OF THE 3-DIMENSIONAL MAXWELL-BLOCH 2-LEVEL LASER SYSTEMS

The three-dimensional Maxwell-Bloch system governs the multi-longitudi nal and transverse mode dynamics of two level wide aperture lasers in an optical ring cavity. The system is hyperbolic in the propagation di rection, and dispersive in the transverse directions due to diffractio n effects. A rich variety of optical patterns and chaos are present in the dynamics. We show the global existence of weak solutions in L(p) (2 less than or equal to p < infinity) spaces of the Maxwell-Bloch sys tem under both absorbing and periodic boundary conditions. The weak so lutions are unique within the class of solutions provided by our regul arization procedure and approach a universal attractor which has only partial smoothing instead of the C-infinity smoothing property found i n early works for the (longitudinal) one-dimensional and (transverse) two-dimensional cases. The idea of the proof makes essential use of bo th the hyperbolicity and dispersivity of the system. In the case of pe riodic boundary condition, our result depends on a conjectural Stricha rtz inequality.