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- W2165880641 abstract "In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of $${L^w(0,T;L^s(mathbb{R}^3))}$$ , where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s − 1). We also show that if $${nabla u in L^w(0,T;L^s(Omega))}$$ with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s − 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, & Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995)." @default.
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- W2165880641 date "2011-09-24" @default.
- W2165880641 modified "2023-09-26" @default.
- W2165880641 title "On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations" @default.
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- W2165880641 doi "https://doi.org/10.1007/s00220-011-1336-4" @default.
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