Dynamical collapse of cylindrical symmetric Dipolar Bose-Einstein condensates

We study the formation of singularities for cylindrical symmetric solutions to the Gross-Pitaevskii equation describing a Dipolar Bose-Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result is a crucial localization property for the fourth power of the Riesz transforms, that we prove by means of the decay properties of the heat kernel associated to the parabolic biharmonic equation, and pointwise estimates for the square of the Riesz transforms. Furthermore, other essential tools are the variational characterization of the Ground State energy, and suitable localized virial identities for cylindrical symmetric functions.