Vol. 3, No. 3, 2021

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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

Vol. 3 (2021), No. 3, 567–594
DOI: 10.2140/paa.2021.3.567

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 L2 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 𝜖-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.

Navier–Stokes equations, $\epsilon$-regularity, regular set, energy concentration, discretely self-similar solutions
Mathematical Subject Classification
Primary: 35Q30, 76D05, 76D03
Received: 20 November 2020
Revised: 8 June 2021
Accepted: 21 August 2021
Published: 24 October 2021
Kyungkeun Kang
Department of Mathematics
Yonsei University
South Korea
Hideyuki Miura
Department of Mathematical and Computing Science
Tokyo Institute of Technology
Tai-Peng Tsai
Department of Mathematics
University of British Columbia
Vancouver, BC