Global Regularity of the 2D HVBK Equations
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Global Regularity of the 2D HVBK Equations Pranava Chaitanya Jayanti1
· Konstantina Trivisa2
Received: 6 April 2020 / Accepted: 22 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Hall–Vinen–Bekharevich–Khalatnikov equations are a macroscopic model of superfluidity at nonzero temperatures. For smooth, compactly supported data, we prove the global well-posedness of strong solutions to these equations in R2 , in the incompressible and isothermal case. The proof utilises a contraction mapping argument to establish local well-posedness for high-regularity data, following which we demonstrate global regularity using an analogue of the Beale–Kato–Majda criterion in this context. In the Appendix, we address the sufficient conditions on a 2D vorticity field, in order to have a finite kinetic energy. Keywords HVBK equations · Superfluids · Global well-posedness · Navier–Stokes equations
1 Introduction Most substances upon isobaric cooling transition from a gas to liquid, before eventually turning into a solid phase (or in some cases, a variety of them). Helium is an exception—at pressures below 25 bars, liquid Helium-4 transforms into a superfluid phase when cooled across the lambda line (approximately 2.17 K).1 This superfluid phase was experimentally discovered (Kapitza 1938; Allen and Misener 1938) over 80 years ago and has since been an important subject of interest to, and investigation by, the physics community. As far as a theoretical explanation of superfluidity goes, 1 Helium-3 also displays a superfluid phase, albeit at a significantly lower temperature (∼2 milliKelvin) due to its fermionic nature. Most experimental research has focused on Helium-4.
Communicated by Pierre Degond.
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Pranava Chaitanya Jayanti [email protected]
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Department of Physics, Centre for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, USA
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Department of Mathematics, Institute for Physical Science and Technology, Centre for Scientific Computation And Mathematical Modeling, University of Maryland, College Park, USA 0123456789().: V,-vol
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Journal of Nonlinear Science
(2021) 31:2
there are several pieces to the puzzle that work in limited ranges of validity; however, a single universal theory that explains everything to reasonable satisfaction continues to elude us. The most well-known models currently in use may be classified in many ways (Barenghi et al. 2014b), the one on the basis of length scales being of interest here. One of the first theories was by Fröhlich (in 1937), proposing a model of order– disorder transition where a fraction of the helium atoms are trapped in a lattice that denotes the ground state, while the remaining atoms are in an excited state. London (1938) modified this theory by suggesting that the ground state could actually be a degenerate Bose gas. Shortly thereafter, Tisza (1938) worked on the details of London’s model, suggesting modifications to match with experimental data. Landau (1941)
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