Nonlinear Schrِdinger equations with coupled Hartree-type terms and rotation

We consider the Cauchy problem for a class of magnetic Schrِdinger system with local and nonlocal nonlinearities. The problem stems from a typical model describing the mean-field dynamics of rotating many-body bosons in a confining trap. We present sufficient conditions which yield global well-posedness or finite time blowup solutions to the system.