  We investigate a micro-scale model of superfluidity derived by Pitaevskii in
1959 to describe the interacting dynamics between the superfluid and normal
fluid phases of Helium-4. The model involves the nonlinear Schr\"odinger
equation (NLS) and the Navier-Stokes equations (NSE), coupled to each other via
a bidirectional nonlinear relaxation mechanism. Depending on the nature of the
nonlinearity in the NLS, we prove global/almost global existence of solutions
to this system in $\mathbb{T}^2$ -- strong in wavefunction and velocity, and
weak in density.
