Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

Abstract

We review some recent results on the long time dynamics of solutions to the Gross-Pitaevskii equation governing non-trapped dipolar Quantum Gases. We describe the asymptotic behaviours 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 heat kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental tool to establish the dynamical features of the solutions to the equation.