The HADES seminar on Tuesday, **May 7th**, will be at **3:30pm** in **Room 748.**

**Speaker:** Pranava Jayanti

**Abstract:** We introduce the physics of superfluidity, including two mathematical models. We begin with a micro-scale description of the interacting dynamics between the superfluid and normal fluid phases of Helium-4 at length scales much smaller than the inter-vortex spacing. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear 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 $\mathbb{T}^n$ (for n=2,3) for a power-type nonlinearity, 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 deriving time-independent a priori estimates. We compare two different approaches: purely energy based, versus a combination of energy estimates and maximal regularity. The results are from recent collaborations with Juhi Jang and Igor Kukavica.

We will also briefly discuss some results pertaining to a macro-scale model known as the HVBK equations, some of which is joint work with Konstantina Trivisa.