Local Existence of Solutions to a Navier–Stokes-Nonlinear-Schrödinger Model of Superfluidity

Abstract

In Pitaevskii (Sov Phys JETP 35(8):282–287, 1959), a micro-scale model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations (NSE). In this article, we show the local existence of solutions—strong in wavefunction and velocity, weak in density—to this system in a smooth bounded domain in 3D, by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim (SIAM J Math Anal 18(1):89–96, 1987) for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE.