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

2021/11/30 by Barker, Tobias, Wang, Wendong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.15444

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 conditions π∈ L^r,∞_tL^s,∞_x \textrmwith \tfrac2r+\tfrac3s=2 \textrmand r∈ (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-2 \textrmwith q∈ (2,3). The second ingredient requires a modification of ideas in Ladyzhenskaya and Seregin's paper, which build upon ideas in Lin, as well as Caffarelli, Kohn and Nirenberg.

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