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

Abstract

In this paper, we investigate systematically the supercritical conditions on the pressure π 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 π satisfies the endpoint scale invariant conditionsπ∈Ltr,∞Lxs,∞with2r+3s=2andr∈(1,∞), 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 π and (ii) the establishment of a convenient ε-regularity criterion involving space-time integrals of|∇v|2|v|q−2withq∈(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].