Ohkitani, K. (2017) Characterization of blowup for the Navier-Stokes equations using vector potentials. AIP Advances, 7 (1). 015211.
Abstract
We characterize a possible blowup for the 3D Navier-Stokes on the basis of dynamical equations for vector potentials A . This is motivated by a known interpolation k A k BMO ≤ k u k L 3 , together with recent mathematical results. First, by working out an inversion formula for singular integrals that appear in the governing equations, we derive a criterion using the nonlinear term of A as R t ∗ 0 k ∂ A ∂t − ν 4 A k L ∞ dt = ∞ for a blowup at t ∗ . Second, for a particular form of a scale-invariant singularity of the nonlinear term we show that the vector potential becomes unbounded in its L ∞ and BMO norms. Using the stream function, we also consider the 2D Navier-Stokes equations to seek an alternative proof of their known global regularity. It is not yet proven that the BMO norm of vector potentials in 3D (or, the stream function in 2D) serve as a blow up criterion in more general cases.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Funding Information: | Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL (EPSRC) EP/N022548/1 |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 01 Feb 2017 10:18 |
Last Modified: | 01 Feb 2017 14:22 |
Published Version: | https://doi.org/10.1063/1.4975406 |
Status: | Published |
Publisher: | AIP Publishing |
Refereed: | Yes |
Identification Number: | 10.1063/1.4975406 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:111101 |