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Abstract

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair $({\bf u},p)$ of suitable weak solutions to the Navier-Stokes equations in $\mathbb R^3\times ]0,\infty [$ the velocity field $\bf u$ satisfies the following property of partial regularity: The velocity $\bf u$ is Lipschitz continuous in a neighbourhood of a point $(x_0,t_0)\in \Omega \times ]0,\infty [$ if $$ \limsup_{R \to 0^+} \frac {1} {R} \int_{Q_R(x_0,t_0)} \biggl|\mathop{\rm curl}{\bf u} \times \frac {\bf u} {|\bf u|} \biggr|^2 \,{\rm d} x\,{\rm d} t \leq\varepsilon _\star $$ for a sufficiently small $\varepsilon _\star>0$.