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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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