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On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. (English) Zbl 0954.35129
The authors consider certain weak solutions of the Navier-Stokes equations in a bounded domain $$\Omega\subset \mathbb{R}^3$$, i.e. \begin{alignedat}{2} v_t-\Delta v+ (v\cdot\nabla)v+\nabla p & =f &&\quad\text{in }\Omega\times (0,T),\\ \text{div }v &= 0 &&\quad\text{in }\Omega\times (0,T),\\ v &= 0 &&\quad\text{on }\partial\Omega\times (0,T),\\ v(0) &= a &&\quad\text{in }\Omega.\end{alignedat} It is assumed that the velocity $$v$$ belongs to $$L^\infty(0,T; L^2(\Omega))\cap L^2(0, T; W^1_2(\Omega))$$, $$p\in L^{{3\over 2}}(\Omega\times (0,T))$$ and that $$(v,p)$$ satisfies a localized energy inequality. The main result of the paper is a partial regularity theorem, which says that if the quantity $\limsup_{R\to 0} {1\over R} \int^{t_0}_{t_0- R^2} \int_{B_R(x_0)}|\nabla v|^2$ is sufficiently small at a point $$(x_0,t_0)\in \Omega\times (0,T)$$, then $$(x,t)\mapsto v(x,t)$$ is Hölder continuous near $$(x_0,t_0)$$. Furthermore, the complement of the points having the above property has parabolic Hausdorff dimension $$\leq 1$$. The proof is based on a blow-up argument combined with a special invariant structure of the equations.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35D10 Regularity of generalized solutions of PDE (MSC2000)
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