ACMD Seminar: Mass Transfer, Energy Dissipation, and Global Solutions in a Micro-scale Model of Superfluidity

Pranava Jayanti
RTPC Assistant Professor, Dept. of Mathematics, University of Southern California

Tuesday, August 20, 2024, 3:00-4:00 PM ET (1:00-2:00 PM MT)

Abstract: 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 Schr\"odinger 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 $\mathbb{T}^d$ $(d=2,3)$ 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 obtaining 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. 

Bio: Pranava completed his Bachelors and Masters degrees in Mechanical Engineering (with a focus on Thermal Engineering, and a minor in Physics) from the Indian Institute of Technology Madras in 2016. Subsequently, he obtained an MS and PhD in physics from the University of Maryland College Park in 2022, with a thesis on the mathematical analysis of different models of superfluidity. He is currently a postdoc in the Department of Mathematics at the University of Southern California. His research interests revolve around the mathematical modeling of physical phenomena, primarily partial differential equations and fluid mechanics.

Host: Stephen Sorokanich

Note: This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

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