Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation

Abstract

We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation −Δs in the critical case of s=32, which satisfy certain local energy inequalities and whose singular sets have a locally finite two-dimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for s=32. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when s=32, the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentration-compactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.