### A direct proof of the Caffarelli-Kohn-Nirenberg theorem

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0\u207a}1/R{\int}_{{Q}_{R}(x\u2080,t\u2080)}|curlu\times u/|u\left|\right|\xb2dxdt\le {\epsilon}_{*}$ for a sufficiently small ${\epsilon}_{*}>0$.