Estimates of the singular set for the Navier-Stokes equations with supercritical assumptions on the pressure

In this paper, we investigate systematically the supercritical conditions on the pressure pi associated to a Navier-Stokes solution v (in three-dimensions), which ensure a reduction in the Hausdorff dimension of the singular set at a first potential blow-up time. As a consequence, we show that if the pressure pi satisfies the endpoint scale invariant conditions pi is an element of Lr,infinity t Ls,infinity x with r2 + s3= 2 and r is an element of(1, infinity), then the Hausdorff dimension of the singular set at a first potential blow-up time is arbitrarily small. This hinges on two ingredients: (i) the proof of a higher integrability result for the Navier-Stokes equations with certain supercritical assumptions on pi and (ii) the establishment of a convenient epsilon-regularity criterion involving space-time integrals of | backward difference v|2|v|q-2 with q is an element of (2, 3). The second ingredient requires a modification of ideas in Ladyzhenskaya and Seregin's paper [21], which build upon ideas in Lin [23], as well as Caffarelli, Kohn and Nirenberg [8].(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

In this paper, the authors systematically investigate the supercritical conditions on the pressure pi associated with a Navier-Stokes solution v in three dimensions. They show that if the pressure pi satisfies certain endpoint scale invariant conditions, then the Hausdorff dimension of the singular set at a first potential blow-up time can be arbitrarily small. The authors establish a higher integrability result for the Navier-Stokes equations and a convenient epsilon-regularity criterion involving space-time integrals.