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.
The results in this talk are from recent collaborations with Juhi Jang and Igor Kukavica.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
