Local weak solutions to a Navier-Stokes-nonlinear-Schrödinger model of superfluidity

Abstract

In a 1959 paper by Pitaevskii, a macroscopic 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 weak solutions 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's 1987 paper 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.

arXiv comment48 pages; added some references in the Introduction section

References in corpus (9)

• Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$
• On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics
• The Quantum Hydrodynamics system in two space dimensions
• Madelung, Gross-Pitaevskii and Korteweg
• On the Gross-Pitaevskii equation for trapped dipolar quantum gases
• Well/ill posedness for the Euler-Korteweg-Poisson system and related problems
• Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples
• Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$
• Local weak solutions to a Navier-Stokes-nonlinear-Schrödinger model of superfluidity

Cited by in corpus (3)

• Local weak solutions to a Navier-Stokes-nonlinear-Schrödinger model of superfluidity
• On the mass transfer in the 3D Pitaevskii model
• Small-data global existence of solutions for the Pitaevskii model of superfluidity