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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space
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by Kyungkeun Kang, Baishun Lai, Chen-Chih Lai and Tai-Peng Tsai PDF
Trans. Amer. Math. Soc. 375 (2022), 6701-6746 Request permission

Abstract:

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.
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Additional Information
  • Kyungkeun Kang
  • Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, South Korea
  • MR Author ID: 625070
  • Email: kkang@yonsei.ac.kr
  • Baishun Lai
  • Affiliation: Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education) and School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, People’s Republic of China
  • Email: laibaishun@hunnu.edu.cn
  • Chen-Chih Lai
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
  • Address at time of publication: Department of Mathematics, Columbia University, New York, New York 10027
  • MR Author ID: 1284128
  • ORCID: 0000-0003-3042-9193
  • Email: cl4205@columbia.edu
  • Tai-Peng Tsai
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
  • MR Author ID: 638612
  • ORCID: 0000-0002-9008-1136
  • Email: ttsai@math.ubc.ca
  • Received by editor(s): July 29, 2021
  • Received by editor(s) in revised form: April 20, 2022
  • Published electronically: July 13, 2022
  • Additional Notes: The research of the first author was partially supported by NRF-2019R1A2C1084685.
    The research of the second author was partially supported by NSFC-11971148.
    The research of the third author was supported by the Simons Foundation Math + X Investigator Award #376319 (Michael I. Weinstein).
    The research of both the third and fourth authors was partially supported by the NSERC grant RGPIN-2018-04137.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 6701-6746
  • MSC (2020): Primary 35Q30, 76D03, 76D05, 76D07
  • DOI: https://doi.org/10.1090/tran/8739
  • MathSciNet review: 4474906