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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.
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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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