Abstract In Jayanti and Trivisa (2022 J. Math. Fluid Mech. 24 46), the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in Pitaevskii (1959 Sov. Phys. JETP 8 282–287) and couples the nonlinear Schrödinger equation and the Navier–Stokes equations. 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.
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Circulation and Energy Theorem Preserving Stochastic Fluids
Abstract Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.
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- Award ID(s):
- 1703997
- PAR ID:
- 10271649
- Date Published:
- Journal Name:
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Volume:
- 150
- Issue:
- 6
- ISSN:
- 0308-2105
- Page Range / eLocation ID:
- 2776 to 2814
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We establish the existence of global pathwise solutions for the stochastic non‐Newtonian incompressible fluid equations in two space dimensions. Moreover, we show that said solutions converge in probability to solutions of the stochastic Navier‐Stokes equations in the appropriate limit. Our approach is based on Galerkin approximations and the theory of martingale solutions.
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Abstract We consider solutions of the Navier‐Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence‐free vector‐valued measure of arbitrary mass supported on a smooth curve. First, we prove global well‐posedness for perturbations of the Oseen vortex column in scaling‐critical spaces. Second, we prove local well‐posedness (in a sense to be made precise) when the filament is a smooth, closed, non‐self‐intersecting curve. Besides their physical interest, these results are the first to give well‐posedness in a neighborhood of large self‐similar solutions of 3d Navier‐Stokes, as well as solutions that are locally approximately self‐similar. © 2023 Wiley Periodicals LLC.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/prm.2019.43
