On the Hausdorff Dimension of the Singular Set in Time for Weak Solutions to the Non-stationary Navier–Stokes Equation on Torus

Abstract

In this note, we investigate the Hausdorff dimension of the possible time singular set of weak solutions to the Navier–Stokes equation on the three dimensional torus under some regularity conditions of Serrin’s type (Arch. RationalMech. Anal., 9, 187–195, 1962). The results in the paper relate the regularity conditions of Serrin’s type to the Hausdorff dimension of the time singular set. More precisely, we prove that if a weak solution u belongs to L ^r(0, T; V _α ) then the $\left (1-\frac {r(2\alpha -1)}{4}\right )$-dimensional Hausdorff measure of the time singular set of u is zero. Here, r is just assumed to be positive. We also establish that if a weak solution u belongs to L ^r(0, T; W ^{1, q}) then the $\left (1-\frac {r(2q-3)}{2q}\right )$-dimensional Hausdorff measure of the time singular set of u is zero. When r = 2, α = 1, or r = 2, q = 2, we recover a result of Leray (Acta Math. 63, 193–248, 1934), Scheffer (Commun. Math. Phys. 55, 97–112, 1977), Foias and Temam (J. Math. Pures Appl. 58, 339–368, 1979), and Temam (Navier–Stokes equations and nonlinear functional analysis. SIAM, Philadelphia, 1995). Our results in some way also relate to the regularity results obtained by Giga (J. Differ. Equ. 62, 186–212, 1986).