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Zbl 1009.76015
Chen, Zhi-Min; Price, W.G.
Blow-up rate estimates for weak solutions of the Navier-Stokes equations.
(English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No.2015, 2625-2642 (2001). ISSN 1364-5021; ISSN 1471-2946/e

Summary: We investigate the interior regularity of Leray weak solutions $u$ of Navier-Stokes equations in a domain $\Omega\subset \bbfR^n$ with $n\ge 3$. It is shown that $u$ is regular in a neighbourhood of a point $(x_0,t_0) \in\Omega \times(0,T)$ if there exist constants $0\le \theta<1$ and small $\varepsilon >0$ such that $\lim_{k\to\infty} \text{ess sup}_{Q_{1/k} (x_0,t_0)}|t-t_0|^{\theta/2} |x-x_0|^{1-\theta} |u(x,t) |<\varepsilon$ with $Q_{1/k} (x_0,t_0)= \{x\in\bbfR^n; |x-x_0|<1/ k\} \times (t_0-1/k^2, t_0+1/k^2)$. If $(x_0,t_0)$ is an irregular point of $u$, there exists a sequence of non-zero measure sets $E_{k_i}\subset Q_{1/k_i} (x_0, t_0)$ for $i=1,2,\dots$, such that the blow-up rate estimate $|u(x,t) |\ge \varepsilon|t-t_0|^{-\theta/2} |x-x_0|^{-1+ \theta}$, $(x,t)\in E_{k_i}$ holds.
MSC 2000:
*76D03 Existence, uniqueness, and regularity theory
35Q30 Stokes and Navier-Stokes equations
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: Lorentz spaces; interior regularity; Leray weak solutions; Navier-Stokes equations; blow-up rate estimate

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