Ohkitani, K. (2017) Analogue of the Cole-Hopf transform for the incompressible Navier-Stokes equations and its application. Journal of Turbulence, 18 (5). pp. 465-479. ISSN 1468-5248
Abstract
We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier–Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. (i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. (ii) This in turn gives an additional characterisation of possible singularities for the Navier–Stokes equations. (iii) Some numerical results for the two-dimensional Navier–Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 Informa UK Limited, trading as Taylor & Francis Group. This is an author produced version of a paper subsequently published in Journal of Turbulence. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Navier-Stokes equations; Cole-Hopf transform; Feynman-Kac formula; Duhamel principle |
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: | 07 Mar 2017 15:30 |
Last Modified: | 13 Jul 2023 10:45 |
Status: | Published |
Publisher: | Taylor & Francis |
Refereed: | Yes |
Identification Number: | 10.1080/14685248.2017.1294758 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:113040 |