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Regular sets and an
$\epsilon$-regularity theorem in terms of initial data for
the Navier–Stokes equations
Kyungkeun Kang, Hideyuki Miura and Tai-Peng Tsai |
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Vol. 3 (2021), No. 3, 567–594
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DOI: 10.2140/paa.2021.3.567
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Abstract |
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We are concerned with the size of the regular set for weak solutions to the Navier–Stokes equations. It is shown that if a weighted norm of initial data is finite, the suitable weak solutions are regular in a set above a space-time hypersurface determined by the degree of the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35:6 (1982), 771–831) in various aspects. Our main tool is an -regularity theorem in terms of initial data, which is of independent interest. As applications, we also study energy concentration near a possible blow-up time and regularity for forward discretely self-similar solutions. |
Keywords
Navier–Stokes equations, $\epsilon$-regularity, regular
set, energy concentration, discretely self-similar
solutions
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Mathematical Subject Classification
Primary: 35Q30, 76D05, 76D03
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Milestones
Received: 20 November 2020
Revised: 8 June 2021
Accepted: 21 August 2021
Published: 24 October 2021
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Authors |
| Kyungkeun Kang | |
| Department of Mathematics Yonsei University Seoul South Korea |
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| Hideyuki Miura | |
| Department of Mathematical and
Computing Science Tokyo Institute of Technology Tokyo Japan |
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| Tai-Peng Tsai | |
| Department of Mathematics University of British Columbia Vancouver, BC Canada |
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