U.S. Quantum Information Science School Micro-scale Types of Models Superfluidity Mathematical Analysis of Models of Superfluids Pranava Chaitanya Jayanti Department of Mathematics, University of Southern California Formerly: Department of Physics, University of Maryland Conversion between normal and superfluid phases with changing temperature. Adapted from [Vinen 2004]. Phase diagram of He-4. “Liquid II” = Superfluid. Adapted from [Young 1975]. D = integral length scale; l = inter-vortex spacing; 𝛏 = vortex core radius. Taken from [Jayanti 2022] Macro-scale • D A Young. Phase diagrams of the elements. Technical report, Lawrence Livermore National Lab, 1975. • W F Vinen. The physics of superfluid helium. Technical report, CERN, 2004. • P C Jayanti. Analysis of models of superfluidity. Doctoral dissertation, University of Maryland, 2022. • P C Jayanti, K Trivisa. Local existence of solutions to a Navier-Stokes-Nonlinear-Schrödinger model of superfluidity. J. Math. Fluid Mech., 2022. • P C Jayanti, K Trivisa. Uniqueness in a Navier-Stokes-Nonlinear-Schrödinger model of superfluidity. Nonlinearity, 2022. • J Jang, P C Jayanti, I Kukavica. Small-data global existence of solutions for the Pitaevskii model of superfluidity. arXiv, 2023. • J Jang, P C Jayanti, I Kukavica. Small-data global existence for the Pitaevskii model of superfluidity in 3D. In prep, 2023. • P C Jayanti, K Trivisa. Global Regularity of the 2D HVBK Equations. Journal of Nonlinear Science, 2021. q Most fluids are described by the Navier-Stokes equations (NSE) q Exhibit viscosity: microscopic roots in the entropic drive for particles to exchange momentum and achieve equilibrium, manifesting as dissipation at macroscopic length scales q Superfluids like Helium-4 are fundamentally inviscid: all the particles move in unison q Bose Einstein condensation below 𝑇! ≈2.17𝐾 for He-4 q Superfluid vorticity – discrete filaments, quantized circulation q He-4 (bosonic) and He-3 (fermionic) both display remarkable properties: critical velocity, film flow, fountain pressure q 𝑇= 0: Pure superfluid; 𝑇> 0: Part of normal liquid condenses into the superfluid q Essential to model and quantify the retarding interactions between phases Pitaevskii 𝜕!𝜓+ 𝜆𝐵𝜓= −1 2𝑖Δ𝜓+ 𝜇 𝑖𝜓"𝜓 𝐵= 1 2 −𝑖∇−𝑢# + 𝜇𝜓" 𝜕!𝜌+ ∇⋅𝑢𝜌= 2𝜆ℜ𝜓𝐵𝜓 𝜕! 𝜌𝑢+ ∇⋅𝜌𝑢⊗𝑢+ ∇𝑞= 𝜈Δ𝑢−𝛼𝜌𝑢−2𝜆ℑ∇𝜓𝐵𝜓 ∇⋅𝑢= 0 Hall-Vinen-Bekarevich-Khalatnikov ∇⋅𝑢$ = 0, ∇⋅𝑢% = 0 𝜕!𝑢$ + 𝑢$ ⋅∇𝑢$ + ∇𝑝$ = 𝜈$Δ𝑢$ + 𝜌% 𝜌𝐹 𝜕!𝑢% + 𝑢% ⋅∇𝑢% + ∇𝑝% = 𝜈%Δ𝑢% −𝜌$ 𝜌𝐹 𝐹= −𝐵 2 𝜔% 𝑢$ −𝑢% + 𝐵& 2 𝜔% 𝑢$ −𝑢% ' q Proposed by Pitaevskii in 1959 to model interacting dynamics between superfluid (nonlinear Schrödinger) and normal fluid (Navier-Stokes) q Order parameter 𝜓 describes superfluid; density 𝜌 and velocity 𝑢 characterize normal fluid q Both phases interact through bidirectional, nonlinear coupling operator 𝐵, consisting of relative kinetic energy and potential self-interaction q Normal fluid velocity taken to be divergence-free: incompressible, inhomogeneous fluid q For small data, global-in-time existence of weak solutions established for both 2D and 3D when 𝑝< 4, and exponentially/polynomially long existence time for 2D when 𝑝≥4 [Jang, Jayanti, Kukavica 2023] q Weak-strong uniqueness of solutions also studied [Jayanti, Trivisa 2022] q Main difficulty in analysis stems from sign-indefinite source in the continuity equation – converts superfluid into normal fluid (on average) q Ignore vortex dynamics and relaxation – continuous vorticity field q Assume both fluids classical, and add a nonlinear coupling q Useful when high density of vortex filaments are aligned (2D) q Studies on instability of superfluid Taylor-Couette flow, and quantum turbulence q Dealing with high frequencies: suppress in 𝑢", subgrid modeling like LES, or artificial viscosity q For 2D, system is globally-in-time well-posed: unique, smooth solutions with continuous dependence on initial data [Jayanti, Trivisa 2021] q Current work is on investigating system for analytic-in-time, Gevrey-in-space solutions (subset