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Zbl 1022.35036
Chen, Zhi-Min; Price, W.G.
Morrey space techniques applied to the interior regularity problem of the Navier-Stokes equations.
(English)
[J] Nonlinearity 14, No.6, 1453-1472 (2001). ISSN 0951-7715; ISSN 1361-6544/e

A new criterion of the interior regularity to the weak solutions of the evolutionary Navier-Stokes equations is formulated in the framework of Morrey spaces. Namely, the following theorem is proved: Let $\frac 72 < q \leq 5$, $(t_0,x_0) \in (0, T) \times \Omega$, $0 < r_0 < \text{dist}(\partial \Omega, x_0)$ and $0< t_0 - r_0^2$ with $\partial \Omega$ the boundary of $\Omega$. Suppose that $u$ is a weak solution of equation $$\partial u/ \partial t - \Delta u + \nabla \cdot (u \otimes u) + \nabla \pi = 0 \quad \text{in}\quad (0,T) \times \Omega$$ $$\text{div} u = 0 \quad \text{in}\quad (0,T) \times \Omega.$$ Then $u$ is regular in $(t_0-r_1^2, t_0] \times B_{r_1}(x_0)$ for $r_1 = r_0/2$ provided that $$\sup r^{1-5/q} \left( \int_{t-r^2}^t \int_{|x-y|< r} |u(s,y)|^q dy ds \right)^{1/q} < \epsilon$$ where the constant $\epsilon$ is sufficiently small and the supremum is taken over all $(t-r^2, t] \times B_r(x) \subset (t_0 - r_0^2, t_0] \times B_{r_0}(x_0).$
[Oldřich John (Praha)]
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory

Keywords: Navier-Stokes equations; interior regularity; Morrey spaces

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