Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states

In this work, we study the semiclassical limit of cubic Nonlinear Schrödinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for data with finite Sobolev regularity whose velocity profiles satisfy a quantum Penrose stability condition. This latter condition is always satisfied for small data (with a smallness condition independent of the semiclassical parameter) both in the focusing and the defocusing case, and for small perturbations of a large class of physically relevant examples in the defocusing case, such as local Maxwellian-like profiles.