Critical thresholds in the semiclassical limit of 2-D rotational Schrödinger equations

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well as the quadratic potential V (x), ∂tU + (U − Ωx⊥) · ∇xU = −ΩU⊥−∇xV , with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross–Pitaevskii equation for Bose–Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection. We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations. Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is quantified in terms of the initial spectral gap associated with the 2× 2 initial velocity gradient, λ2(0)−λ1(0), λj(0) = λj(∇xU0) as well as the initial divergence, divx(U0). We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the deformed flow map. Mathematics Subject Classification (2000). Primary 35Q40; Secondary 35B30.