Dynamics of Solutions to the Gross-Pitaevskii Equation Describing Dipolar Bose-Einstein Condensates

We review some recent results on the long-time dynamics of solutions to the Gross-Pitaevskii equation (GPE) governing non-trapped dipolar quantum gases. We describe the asymptotic behaviors of solutions for different initial configurations of the initial datum in the energy space, specifically for data below, above, and at the mass-energy threshold. We revisit some properties of powers of the Riesz transforms by means of the decay properties of the integral kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental role in establishing the dynamical features of the solutions to the studied GPE.