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New sufficient conditions for regularity of solutions to the Navier-Stokes equations. (English) Zbl 1039.35077
Summary: We show that a weak solution $$u$$ to the 3D Navier-Stokes equations belongs to the space $$L^2(0,T;L^\infty({\mathbb R}^3))$$ if one of the following three conditions holds:
(i) $\int^T_0(\sup_{x_0\in{\mathbb R}^3}\int_{{\mathbb R}^3}(|x-x_0|^\alpha u)^qdx)^{s/q}dt<+\infty$ for some $$\alpha\in\mathbb R$$ and $$q>1$$, with $$1/s+3/2q=1/2-\alpha/2,\;\alpha+1\geq0$$ and $$-1<\alpha+3/q<1$$;
(ii) There exists $$r_0>0$$ such that $\epsilon=\sup_{t\in[0,T]}\sup_{x_0\in {\mathbb R}^3}\int_{B_{r_0}(x_0)}|x-x_0|^{-1}|u|^2dx$ is sufficiently small;
(iii) There exists $$r_0>0$$ such that $\int^T_0\sup_{x_0\in{\mathbb R}^3}\int_{B_{r_0}(x_0)}|x-x_0|^{-1}|\nabla u|^2dxdt<+\infty.$

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