Vol. 3, No. 3, 2021

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Navier–Stokes regularity criteria in sum spaces

Evan Miller

Vol. 3 (2021), No. 3, 527–566
DOI: 10.2140/paa.2021.3.527

We will consider regularity criteria for the Navier–Stokes equation in mixed Lebesgue sum spaces. In particular, we will prove regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix, in the sum space of two scale-critical spaces. This represents a significant step forward, because each sum-space regularity criterion covers a whole family of scale-critical regularity criteria in a single estimate. In order to show this, we will also prove a new inclusion and inequality for sum spaces in families of mixed Lebesgue spaces with a scale invariance that is also of independent interest.

Mathematical Subject Classification
Primary: 35Q30
Received: 26 September 2020
Revised: 4 June 2021
Accepted: 21 August 2021
Published: 24 October 2021
Evan Miller
Department of Mathematics and Statistics
McMaster University
Hamilton, ON