  We examine a micro-scale model of superfluidity derived by Pitaevskii in 1959
which describes the interacting dynamics between superfluid He-4 and its normal
fluid phase. This system consists of the nonlinear Schr\"odinger equation and
the incompressible, inhomogeneous Navier-Stokes equations, coupled to each
other via a bidirectional nonlinear relaxation mechanism. The coupling permits
mass/momentum/energy transfer between the phases, and accounts for the
conversion of superfluid into normal fluid. We prove the existence of weak
solutions in $\mathbb{T}^3$ for a power-type nonlinearity, beginning from small
initial data. The main challenge is to control the inter-phase mass transfer in
order to ensure the strict positivity of the normal fluid density, while
obtaining time-independent a priori estimates.
