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We study a dissipative variant of the Gross-Pitaevskii equation withrotation. The model contains a nonlocal, nonlinear term that forces theconservation of $L^2$-norm of solutions. We are motivated by several physicalexperiments and numerical simulations studying the formation of vortices inBose-Einstein condensates. We show local and global well-posedness of thismodel and investigate the asymptotic behavior of its solutions. In the linearcase, the solution asymptotically tends to the eigenspace associated with thesmallest eigenvalue in the decomposition of the initial datum. In the nonlinearcase, we obtain weak convergence to a stationary state. Moreover, for initialenergies in a specific range, we prove strong asymptotic stability of groundstate solutions.

Existence and Large Time Behavior for a Dissipative Variant of the Rotational NLS Equation