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. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional non-linear 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 solutions in 𝕋d (d=2,3) for a power-type non-linearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time.
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. We present two different approaches (purely energy based, versus a combination of energy estimates and maximal regularity) based on the dimension.
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