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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.
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Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier–Stokes equations on the torus
Abstract We consider the three-dimensional Navier–Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier–Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity.
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- Award ID(s):
- 2307638
- PAR ID:
- 10545781
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Monatshefte für Mathematik
- Volume:
- 206
- Issue:
- 4
- ISSN:
- 0026-9255
- Format(s):
- Medium: X Size: p. 781-795
- Size(s):
- p. 781-795
- Sponsoring Org:
- National Science Foundation
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