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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 $π$ 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 $$π\in L^{r,\infty}_{t}L^{s,\infty}_{x}\quad\textrm{with}\,\,\tfrac{2}{r}+\tfrac{3}{s}=2\,\,\textrm{and}\,\,r\in (1,\infty),$$ 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 $π$ and (ii) the establishment of a convenient $\varepsilon$- regularity criterion involving space-time integrals of $$|\nabla v|^2|v|^{q-2}\,\,\,\textrm{with}\,\,q\in (2,3). $$ 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.