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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., Comment: 34 pages. arXiv admin note: text overlap with arXiv:2106.04659

This thesis deals with the rigorous analysis of two models of superfluidity. One of them is a macro-scale description of the interacting dynamics of a mixture of superfluid Helium and normal Helium. The equations used are modifications of the incompressible Navier-Stokes equations in 2D, with a nonlinear \textit{mutual friction} that couples the two fluids. We show global well-posedness of strong solutions (with high-regularity data) to this model, by proving a Beale-Kato-Majda-type condition. This work has been published in the Journal of Nonlinear Science. \\ Next, we study a micro-scale model (the ``Pitaevskii’’ model) of superfluid-normal fluid interactions, derived by Lev Pitaevskii in 1959. This involves the nonlinear Schr\"odinger equation and incompressible inhomogeneous Navier-Stokes equations. Mass and momentum exchange between the two fluids is mediated through a nonlinear and bidirectional coupling. We establish the existence of local solutions (strong in wavefunction and velocity, weak in density) that satisfy an energy equality. The analysis of this model has been published in the Journal of Mathematical Fluid Mechanics. \\ Finally, we prove a weak-strong type uniqueness theorem for the solutions of the Pitaevskii model. We begin by arguing that the standard weak-strong uniqueness argument does not seem to work in the case of weak solutions whose regularity is governed purely by the energy balance equation, even if the strong solution is as smooth as one wishes. Thus, we are forced to consider slightly less weak solutions obtained from a higher-order energy bound. Owing to their better regularity, we can compare them to \textit{moderate} solutions $-$ which are rougher than conventional strong solutions used for this purpose $-$ and establish a \textit{weak-moderate uniqueness} theorem. Relative to the solutions actually constructed in the earlier part of this thesis, only some of the regularity properties are used, allowing room for improved existence theorems in the future, while maintaining compatible uniqueness results. The uniqueness results have been accepted for publication in Nonlinearity.

In a previous paper [Jayanti, P.C., Trivisa, K. Local Existence of Solutions to a Navier-Stokes-Nonlinear-Schrödinger Model of Superfluidity. J. Math. Fluid Mech. 24, 46 (2022)], the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived by Pitaevskii in 1959 and couples the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE). In this article, we prove a weak-strong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results., 20 pages. Major revisions from previous versions including improved estimates and the removal of assumptions (small data/low energy/short time) for uniqueness

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 Schr��dinger 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., 48 pages; added some references in the Introduction section

If two probability density functions (PDFs) have values for their first $n$ moments which are quite close to each other (upper bounds of their differences are known), can it be expected that the PDFs themselves are very similar? Shown below is an algorithm to quantitatively estimate this "similarity" between the given PDFs, depending on how many moments one has information about. This method involves the concept of functions behaving "similarly" at certain "length scales", which is also precisely defined. This technique could find use in data analysis, to compare a data set with a PDF or another data set, without having to fit a functional form to the data.