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 $\pi$ 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 $$\pi \in L_t^{r,\infty }{L}_x^{s,\infty }\text{with}\frac{2}{r}+\frac{3}{s}=2\text{and}r\in (1,\infty ),$$ 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 $$|\nabla v{|}^2|v{|}^{q-2}\text{with}q\in (2,3).$$ 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.

arXiv commentThis is a substantial revision, an author has been added and the title has changed. Now, we systematically treat general supercritical assumptions on the pressure. In the endpoint scale-invariant cases for the pressure, we obtain a sharper estimate of the Hausdorff dimension of the singular set. 22 pages, 1 figure