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On the Hausdorff dimension of the singular set in time for weak solutions to the non-stationary Navier-Stokes equation on torus. (English) Zbl 1326.35242
In this paper, the authors are concerned with 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. The initial value problem for the non-stationary Navier-Stokes equations is considered. The results in the paper are of follwing type. It is proved that if a weak solution $$u$$ is an element of $$L^{r}(0, T ; V_{\alpha})$$ then the $$\left( 1 - \frac{r (2 \alpha - 1)}{4}\right)$$-dimensional Hausdorff measure of the corresponding time singular set of $$u$$ is zero. Furthermore, the authors prove 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 function $$u$$ is again zero. For particular values of $$r, \alpha, q$$ the authors recover results by J. Leray [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] and by R. Temam [Navier-Stokes equations and nonlinear functional analysis. 2nd ed. Philadelphia, PA: SIAM (1995; Zbl 0833.35110)]. The paper is self-contained and comprehensive.
The bibliography contains 18 items.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35D30 Weak solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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##### References:
 [1] Beirão de Veiga, H, A new regularity class for the Navier-Stokes equations in \^{}{$$n$$}, Chin. Ann. Math. Ser. B, 16, 407-412, (1995) · Zbl 0837.35111 [2] Caffarelli, L; Kohn, R; Nirenberg, L, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Math. Appl., 35, 771-831, (1982) · Zbl 0509.35067 [3] Foias, C; Temam, R, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, 339-368, (1979) · Zbl 0454.35073 [4] Giga, Y, Solutions of semilinear parabolic equations in lp and regularity of weak solutions of the Navier-Stokes systems, J. Differ. Equ., 62, 186-212, (1986) · Zbl 0577.35058 [5] Hopf, E, Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4, 213-231, (1951) · Zbl 0042.10604 [6] Hou, TY; Lei, Z, On the partial regularity of a 3D model of the Navier-Stokes equations, Commun. Math. Phys., 287, 589-612, (2009) · Zbl 1176.35130 [7] Ladyzhenskaya, OA, Solution “in the large” of the nonstationary value problem for the Navier-Stokes system with two space variables, Commun. Pure Appl. Math., 12, 427-433, (1959) · Zbl 0103.19502 [8] Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1963) · Zbl 0121.42701 [9] Leray, J, Essai sur LES mouvements d’un liquide visquex emplissant l’espace, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05 [10] Robinson, JC; Sadowski, W, On the dimension of the singular set of solutions to the Navier-Stokes equations, Commun. Math. Phys., 309, 497-506, (2012) · Zbl 1233.35162 [11] Scheffer, V; Temam, R (ed.), Turbulence and Hausdorff dimension, in turbulence and Navier-Stokes equations, 94-112, (1976), New York [12] Scheffer, V, Hausdorff measure and the Navier-Stokes equations, Commun. Math. Phys., 55, 97-112, (1977) · Zbl 0357.35071 [13] Serrin, J, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9, 187-195, (1962) · Zbl 0106.18302 [14] Serrin, J; Langer, RE (ed.), The initial value problem for the Navier-Stokes equations, 69-98, (1963), Madison [15] Sohr, H.: The Navier-Stokes equations: an elementary functional Analytic Approach. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel (2001) · Zbl 1388.35001 [16] Struwe, M, On partial regularity results for the Navier-Stokes equations, Commun. Pure Appl. Math., 41, 437-458, (1988) · Zbl 0632.76034 [17] Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland (1979) · Zbl 0426.35003 [18] Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics. 2nd edn., Society for Industrial and Applied Mathematics (SIAM), vol. 66, Philadelphia (1995) · Zbl 0833.35110
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